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Matheology

§ 001 A matheologian is a man, or, in rare cases, a woman, who in that nobody can think, except, perhaps, a God, or, in rare cases, a Goddess.

§ 002 Can the of God be proved from ?

Gödel proved the existence of God in a relatively complicated way using the positive and negative properties introduced by Leibniz and the axiomatic method ("the axiomatic method is very powerful", he said with a faint smile). http://www.stats.uwaterloo.ca/~cgsmall/ontology.html http://userpages.uni-koblenz.de/~beckert/Lehre/Seminar-LogikaufAbwegen/graf_folien.pdf

Couldn't the following simple way be more effective? 1) The of real is uncountable. 2) Humans can only identify countably many words. 3) Humans cannot distinguish what they cannot identify. 4) Humans cannot well-order what they cannot distinguish. 5) numbers can be well-ordered. 6) If this is true, then there must be a with higher capacities than any human. QED

[I K Rus: "Can the existence of god be proved from mathematics?", .stackexchange, May 1, 2012] http://philosophy.stackexchange.com/questions/2702/can-the-existence-of-god-be-proved-from- mathematics

The appending discussion is not electrifying for . But a similar question had been asked by I K Rus in MathOverflow. There the following more educational discussion occurred (unfortunately it is no longer accessible there).

(3) breaks down, because although I can't identify (i.e. literally "list") every real between 0 and 1, if I am given any two real numbers in that interval then I can distinguish them. – C GERIG

If you are given two numbers, then both can be given, i.e., belong to the of finite expressions. – I K RUS

I voted down to close as "subjective and argumentative". Claiming that the well-ordering axiom implies that someone can order the reals is really inane, in my opinion. – ANGELO

I agree. It is really inane. But most mathematicians don't even know that this is inane. We should teach them: It is really inane to believe that all real numbers "exist" unless God has a list of them. – I K RUS

God is not the of proof. Either you believe or not, but this is only a of faith. It would be too simple if a proof of existence or non-existence existed. We should not have any . – D SERRE

God is the subject of Gödel's proof. God is the subject of my proof. And I am very proud that I have devised a proof that can be understood by a cobblers apprentice (as Euler requested). That will pave my way into the paradise. We know, without God there is no paradise, not even Hilbert's. You rightfully remark, "we should not have any choice." And we have no choice - unless we have the axiom of choice. Now I will no longer respond to questions and comments and will withdraw into my hermitage. Bless you God. – I K RUS

Although I agree with the closing of your question, thanks for bringing up that webpage - it is interesting and useful. Knowledge can come from many sources. – F GOLDBERG

Yes, but unfortunately in MathOverflow it seems to be not always appreciated. This instructive question and discussion have been closed as spam and deleted immediately.

§ 003 Is the analysis as taught in universities in fact the analysis of definable numbers?

In October 2010 this question had been asked in MathOverflow by user ANIXX. The following is substantially shortened. For full text see here: http://mathoverflow.net/questions/44102/is-the-analysis-as-taught-in-universities-in-fact-the- analysis-of-definable-numbe All numbers are divided into two classes: those which can be unambiguously defined by a limited set of their properties (definable) and such that for any limited set of their properties there is at least one other number which also satisfies all these properties (undefinable). It is evident that since the number of properties is countable, the set of definable numbers is countable. So the set of undefinable numbers forms a continuum. ... But the main question that bothered me was that the analysis course we received heavily relied on constructs such as "let's a to be a number that...", "for each s in interval..." etc. These seemed to heavily exploit the properties of definable numbers and as such one can expect the theorems of analysis to be correct only on the set of definable numbers. ... – ANIXX

The naive account continues by saying that since there are only countably many such ϕ, but uncountably many reals, there must be reals that we cannot describe or define. But this line of reasoning is flawed in a number of ways and ultimately incorrect. The basic problem is that the naive of definable number does not actually succeed as a definition. ... I am taking this opportunity to add a link to my very recent paper "Pointwise Definable Models of ", J. D. Hamkins, D. Linetsky, J. Reitz, which explains some of these definability issues more fully. – J D HAMKINS

As I understand you say it can be postulated in ZFC that undefinable numbers simply do not exist. – ANIXX

No, this is not what Joel was saying. He did not say that it is consistent to "postulate in ZFC that undefinable numbers do not exist". What he was saying was that ZFC cannot even express the "is definable in ZFC". – A BAUER

The Preprint by J D HAMKINS et al. http://de.arxiv.org/abs/1105.4597 contains the following phrases, starting smugly:

One occasionally hears the argument—let us call it the math-tea argument, for perhaps it is heard at a good math tea—that there must be real numbers that we cannot describe or define, because there are are only countably many , but uncountably many reals. Does it withstand scrutiny? Question 1. Is it consistent with the axioms of set theory that every real is definable in the language of set theory without parameters? The answer is Yes. Indeed, much more is true: if the ZFC axioms of set theory are consistent, then there are models of ZFC in which every , including every , every function on the reals, every set of reals, every topological space, every ordinal and so on, is uniquely definable without parameters. Inside such a universe, the math-tea argument comes ultimately to a false conclusion. The number of descriptions is countable, and there is a one-to-one function mapping definitions to the objects they define in a pointwise definable model (so all such models are countable).

Okay lets take standard analysis. It follows that the number of reals is uncountable (inside this model) while the number of definable numbers is countable. How can we be confident the analysis theorems that employ definablility actually true for all reals? – And yes, I do not say undefinable numbers do not exist. Their existence follows from Axiom of choice and in theory we can uniquely define each undefinable number by specifying infinite number of its properties. The problem is that the theorems of analysis as in universities sufficiently rely on the properties of definable numbers. – ANIXX

A r is a number that can be defined, i.e., r can be identified and communicated by a finite sequence of bits in real life, just where mathematics takes place. This makes the set of simultaneously (in a given language) definable numbers countable. Therefore all real numbers that can appear in the language of belong to a countable set. Independent of real-life conditions it is impossible to distinguish, in the universe of ZFC or elsewhere, real numbers by infinite sequences of bits. This claim is proven by the possibility to construct all infinite sequences of digits by means of a countable set of infinite sequences of digits as follows: Enumerate all nodes ai of an infinite binary tree and map them on infinite paths such that ai œ pi. There is no further restriction. The mapping need not be injective. Then construct from this countable set of paths another binary tree. Mathematical analysis is not able to discern which paths were used for construction. This shows that outside of a platonist ZFC-universe there are not uncountably many real numbers. Real numbers created by Cantor-lists are not defined unless the Cantor-list is well- defined, i.e., every entry of the list is known. That requires a Cantor-list constructed by a finite definition. But there are only countably many finite definitions of Cantor-lists. The existing real numbers of analysis cannot be listed. But that does not make their set larger than any countable set. – USER

This last answer, however, enjoyed only a very short lifetime before it has been locked and deleted. (Also the original question had been closed very soon. It has only be reopened by intervention of J D HAMKINS.)

§ 004 On March 10, 2009 I asked in FOM (an automated e-mail list for discussing foundations of mathematics): Who was the first to accept undefinable individuals in mathematics? http://www.cs.nyu.edu/pipermail/fom/2009-March/013464.html Until the end of the nineteenth century mathematicians dealt with definable numbers only. This was the most natural thing in the world. An example can be found in a letter from Cantor to Hilbert, dated August 6, 1906:

"Infinite definitions (that do not happen in finite ) are non-things. If Koenigs theorem was correct, according to which all finitely definable numbers form a set of ¡0, this would imply that the whole continuum was countable, and that is certainly false."

Today we know that Cantor was wrong and that an uncountable continuum implies the existence of undefinable numbers. {{This was Wolfgang in sheep's clothing. In fact I should have written: The implied necessity of undefinable numbers proves the non-existence of uncountable sets, because numbers are definitions and undefinable definitions are nonsense. But then I could have spared sending off my text and could have glued it above my desk instead.}} Who was the first to deliberately accept undefinable individuals like real numbers in mathematics?

This question appeared the same day (obviously someone hadn't paid enough attention) and raised some resonance. Details can be found in http://www.cs.nyu.edu/pipermail/fom/2009-March/

But, in any case, definable in what language? ... Surely Cantor was wrong only in the sense that he didn't point out that the notion of definability cannot be absolute, but depends upon the language. - B TAIT

Unfortunately all my intended answers like the following were "deemed inappropriate by the moderator":

One fact is independent of any model: Every calculation is limited to a finite amount of bits. Every set the elements of which can be treated as individuals is countable. The set of finite words over a finite alphabet is countable. The set of meanings of these words, i.e., the set of languages, is countable. The set of finite alphabets is countable. The Cartesian product of these, and possibly some further features, is countable. These facts are independent of the physical model and independent of the logical stand point.

The repeated rejections annoyed me, and I directed a complaint to the list administrator.

Is this referee a university teacher who is supposed to know something and to explain what he knows?

That, in turn, obviously annoyed the referee. He wrote to the moderator: Mueckenheim adds to his previously rejected nonsense the false assertion that the Cartesian product of the set of finite alphabets is countable. As for university teachers being supposed to explain what we know, I've wasted far too much time trying to explain mathematics (or even common sense) to people who insist on some incorrect "proof." I don't consider it my duty or yours to waste more time in that direction. Best regards, ...

Nevertheless it is fact that the Cartesian product of finitely many countable sets is countable - even if the referee cannot understand that. Nothing hinders the scientific progress as much as the presumptuousness of incompetent referees. So I addressed the moderators again, and the following discussion evolved:

Referee to FOM: He's right that some statements about definability don't need details of the language, but I think he's wrong to infer (in his first sentence) "an absolute of undefinability."

WM: I do not infer an absolute meanig about undefinability. But I infer that only countably many real numbers can be defined.

Referee to FOM: Does he perhaps think there's a particular real number that is undefinable in any (countable) language?

WM: That's a tricky question. Of course I cannot define an undefinable number. ... I would need to make an infinite sequence of decisions 0 or 1 in order to determine the binary representation of a real number without a finite defining equation. Of course that is impossible for me and for ¡0 ¡0 ¡0 anyone else. Anyhow, if there are 2 real numbers, then there are 2 - ¡0 = 2 of such undefinable real numbers.

After some exchange of arguments I wrote:

Dear Sirs, I am surprised that some of you seem to be believe that there are uncountably many languages or that a Cartesian product of three countable sets could be uncountable. If you understand that a language including thesaurus is nothing but a finite definition, then you will agree that all definitions including all languages and all theorems, proofs, numbers, symbols etc. are present in the following list of all finite words: 0 1 00 01 10 11 000 ... This list is the list of everything. Of course some of its words may have different meanings, according to the language applied. (Definitions of languages can be found in some later parts of the list.) The only item that is missing is a diagonal word. It is easy to see that a diagonal word cannot be constructed. And even if it could, it would not mean anything because all meaningful words have to be finite. If however you are really unable to understand that this simple logical chain is as justified as what you may think {{a purely polite phrase; what they think is not in the least justified}}, then I have to consider FOM as an abbreviation of Fools Of Matheology instead. And I would kindly ask you to delete me from your list as it would be only waste of time to read what you write. Regards, WM

After all I received an email with the subject: "You have been unsubscribed from the FOM mailing list" without love and kisses and any further text. The moderators were disgruntled. I can understand that. I would be disgruntled too if someone claimed that my life's work was humbug - and if he could prove that.

§ 005 In order to save the of the set of uncountably many real numbers, these numbers must be distinct by some . As there are not enough finite properties (definitions, , strings of digits), it is assumed in matheology, that the elements of the set — carry infinite names such that they can be distinguished in or by a God or, in rare cases, by a Goddess. But that idea can be contradicted by the Binary Tree.

A due question has been asked in MathOverflow by ANO NYMUS on May 5, 2012: A constructivist’s puzzling argument: I enumerate all nodes ai of an infinite binary tree and map them on infinite paths pi such that ai œ pi. There is no further restriction. The mapping need not be injective (and cannot be surjective). But I don’t tell you which paths pi I have chosen. Then I construct from this countable set of paths another binary tree. You are not able to find out which paths I have used. You cannot even determine whether or not I used the path of the 1/3. That proves that real numbers cannot be defined by digits alone, contrary to Cantor’s diagonal argument, where the diagonal number is defined by digits, in that it differs from every other number by at least one digit. Result: Real numbers need finite definitions. But there are only countably many. – ANO NYMUS

He does not distinguish between "infinite" and "arbitrarily large finite" – B KJOS-HANSSEN

Just this is the question: How can "infinite" and "arbitrary large" be distinguished other than by finite definitions? – ANO NYMUS

Wait, isn't 1/3 defined by a digit? Or if you prefer a decimal system, isn't 1/10 defined by a digit? Hooray, these are not real numbers! – ASAF KARAGILA

These are real numbers, defined by three and four symbols, respectively. I would really prefer an answer instead of polemics. – ANO NYMUS

No matter how finitely many digits 3.14159265358979 I give you, you can't be sure what real number those are the first digits of. From this it seems he concludes that numbers are not defined by digits alone. The number π has a finitely long definition (involving circumferences and diameters etc.). It's just that the definition does not consist in listing some of the digits. – B KJOS-HANSSEN

The argument is Mückenheimian philosophy, so I'm pretty sure ano nymus is yet another sockpuppet of WM: hs-augsburg.de/~mueckenh/MR/.pdf – M GREINECKER

{{My argument is not philosophical but purely mathematical. If somebody claims that uncountably many real numbers can be distinguished by infinite strings of bits, but is unable to discern which of countably many infinite strings make up the complete infinite Binary Tree, then it's about time that he should try to get accustomed step by step to the faint idea that he was dreaming. I can only encourage everyone to look into the recommended document.}}

Obviously π has many finite definitions. One of them is "π". – ANO NYMUS

You only think this is a real question. It is not. – M GREINECKER

That's a very good argument!? I must confess, I think it is a real question, and that puzzles me. From the answers I got here and from the "closing as spam", in particular as one of the closers obviously has not understood what is asked for at all, I have been shocked. If you do not think it is a real question, you should say why. And why should my question be spam??? – ANO NYMUS

After few hours this question and discussion have been closed as spam and immediately deleted. An answer could not be given, as had to be expected.

§ 006 Is there a well-ordering of the reals, measurable or not?

This question was asked by a user of Math.StackExchange in April 2012: I just stumbled on these two claims: "Nice" well-orderings of the reals and, in the answer, "no well-ordering of the reals is Lebesgue measurable". And I am surprised. Is there a well-ordering of the reals after all, measurable or not? - USER

Assuming the axiom of choice, yes. Any well ordering of the reals, as a subset of the plane, is not measurable. - ASAF KARAGILA

I am only asking whether there is a well-ordering of the reals. I think I remember that somebody proved it impossible. But I am not sure. - USER

It depends entirely on your set theory. If you assume the axiom of choice, then there is a well- ordering of every set, and in particular of the reals. - BRIAN M SCOTT

It seems you dispute (incorrectly) that uncountable sets can be well-ordered. {{No, no, no. I swear by Cantor, Zermelo and Gödel that I truely believe in matheology. (Otherwise this question and discussion certainly would have been closed and deleted immediately as is customary in media run by religious fanatics or dictatorial regimes despising human rights and freedom of word.)}} ... You do not even need axiom of choice. - SDCVVC

I am curious how the real numbers can be well-ordered without attaching an expression to each of them. And that makes me doubt a real well-ordering, if it is not given by a formula that allows to test it. A proof based on axioms would not be useful. For instance if we use the axiom "every sum of two natural numbers is even", we can prove that result. Nevertheless ... - USER

From the axioms of ZFC we can prove the existence of a well ordering of any set. In particular the set of real numbers. ... Such well ordering, though, is not nice in the sense that any formula defining it would be rather complicated (compared to other sets which we know, like open sets). - ASAF KARAGILA

You say: "any formula defining it would be rather complicated". So it is not excluded that there could be a formula defining it? Then the proof that I read is wrong? - USER

1 In the there is a formula of complexity Σ 2 defining a well ordering of the real numbers. The same formula, in larger universes might not define a well-ordering at all. - ASAF KARAGILA

I am talking about real numbers as they are defined in a high school math course. Is it possible to well-order them by a formula in our universe? Can you give a formula where the relative positions can be read? - USER

The real numbers are not defined in high school. If you have seen a definition of the real numbers, you're probably one of the lucky few. Regardless of that we can define these numbers as sets in set theory. I should also explicitly say that when I say formula I am not talking about "x3 - ey" sort of formula but rather something very very different. - ASAF KARAGILA

I have learned equivalence classes of Cauchy-sequences, Dedekind-cuts and interval-nesting. Only the definition of Weierstrass is unknown to me. I know that the rational numbers can be put in bijection with the naturals and the real numbers cannot. Further I know that all finite words belong to a countable set. That's why I wonder how reals are well ordered without having a for each one. I would be interested in seeing a formula which does that. - USER

Historically before the axiom of choice and the work with began, mathematics dealt with things "we can name" for the most part. After the axiom of choice was formulated, and infinities began to take form and shape, we began researching things which we can deduce their existence but not name. In a way this is similar to black holes (but this analogy does not go very far). We cannot see them, we can only prove their existence due to some laws of which we believe hold. Same here, we cannot write down a well-ordering but we can prove it exists. - ASAF KARAGILA

Black holes are a nice example. But to prove, that numbers (which cannot be well-ordered, because they cannot be named), can be well-ordered nevertheless, is really a joke. Are people get paid by tax payers for that "research"? And do they know what you are doing? - USER

Similarly, note that you cannot really name every rational number (since your brain is limited in size and ability to comprehend longer and longer words) but you can still prove the infinitude of rational numbers between zero and one. - ASAF KARAGILA

We can in principle name each one. That is quite different with the real numbers. So this example of yours fails. - USER

You obviously misunderstand what does "prove" mean. To prove means to deduce by a finite of steps, from certain assertion, a certain conclusion. To prove something does not mean "finding a name for it". - ASAF KARAGILA

Mathematicians have proven that, assuming "mathematics" (i.e., ZF) {{that is the usual usurpatory abuse of the word "mathematics" by matheologians}} is consistent, if you want to assume "— can be well ordered", you can without adding any . Somewhat surprisingly, if you want to assume "— cannot be well ordered", this is equally ok - no contradictions will be added. Practically, this means you cannot prove "— can be well ordered" or else you wouldn't be able to assume it can't be without . Likewise, you cannot prove "— cannot be well ordered", for the same . - J DEVITO

Your comment on taxes betrays a rather deep misunderstanding of the matters which you are attempting to discuss, and a rather large and misguided axe to grind. This is the place for neither. - A MAGIDIN

User did not answer, and without any authorization or consent from my side this discussion has been faked in that "user" has been replaced by my name. http://math.stackexchange.com/questions/137657/is-there-a-well-ordering-of-the-reals- measurable-or-not This is a violation not only of copy right and personality laws. But taking legal action would not be worthwhile. So let me deputize for the user, taking the advantage of answering the last three contributions as kind of co-author:

"Proving" in matheology seems to boil down to results like this: "If cowshit tastes like honey, then we should try horse droppings." But I am not interested in such "proofs". I am asking without fuss or quibble: Would you try horse droppings?

§ 007 In Math.StackExchange the following question was put by a user: What is the difference between the made up case A and the real case B? A) We use the axiom: Every sum of two perfect numbers is even. Based on the number 6 we prove from this axiom that there is no odd perfect number. In case an odd perfect number could be found, this axiom must be abolished. B) We use the axiom of choice and prove from it that every set of real numbers can be well- ordered. From the fact that there are not enough names available and that ordering of numbers without names is impossible, we can conclude that the axiom must be abolished. So what is the difference? Why is has this axiom not yet been abolished? To prove something does not mean "finding a name for it". But to prove that it is possible to order every name (and real numbers belong to the set of names), does mean that it is possible to have every name. - USER

Very soon this question received some amazing answers:

Could you spell out what you mean by "ordering of numbers without names is impossible"? That seems to be the crucial point. In the usual treatment of the subject, this is not true. – JORIKI

Your claim that "ordering numbers without names is impossible" is simply unjustified. – M SUÁREZ-ALVAREZ

Likewise "real numbers belong to the set of names". Numbers are not names. – C EAGLE

The axiom you need to abolish is "an immaterial object cannot be put in any well-ordering unless you can refer to it". – SDCVVC

{{But is the reverse true without axiom? I think, matheologians should add as the eleventh axiom to ZFC: 10. Ordering does not require knowledge of the ordered elements. At least the axiom of extensionality

Given any set A and any set B. If for every set C, C is a member of A C is a member of B, then A is equal to B. should be reformed slightly, because we cannot determine whether C is a member of a A unless C "is given", i.e., C is definable in a finite time by a signal of finite (although arbitrarily high) bit- rate.}}

You must also object to our summing numbers which we cannot name, and multiplying them. – M SUÁREZ-ALVAREZ

{{I have never summed numbers which I cannot name - at least in principle and when disregarding MatheRealism.}}

In any case, please remember that this is not a discussion site: you ask questions and others answer, but if you want to debate the answers you should find another venue. – M SUÁREZ- ALVAREZ

{{Done. Too much discussion is bad for any religious branch.}}

How do you hate people that you don't know as individuals? Yet so many people in the world hate to the bone groups of other people for their sex, the color of their skin, their religion, their personal beliefs, their sexual preference, their musical preference, etc. etc. – ASAF KARAGILA

I think one cannot hate unknown people, even less one can put them in an order. One may dislike certain habits of a group of people, as one may know certain properties of a set of numbers like the reals. But it is a bit of ridiculous to defend the obviously self-contradictory idea of unthinkable thoughts and unspeakable words by such analogies. – B ZARKIN

OP's claim in the question is that it is impossible. If it has to stand, it should be proven. It is fallacious to claim it is "apparently impossible" and switch the burden of the proof. That's not how mathematics works. ... mathematics uses ZF(C) everywhere by default, and failure of intuition is a problem of a human, not a problem of mathematics. –SDCVVC

{{So it belongs to the realm of "intuition" to believe that I cannot order what I cannot distinguish. In this manner matheology can be defended till the end of days. Then every contradiction can be avoided by observing that some manipulations have been applied that are not axiomatized in ZFC.}}

It is not intuition to conclude that 10 objects cannot be distinguished by 5 labels. But it is intuition, and in my opinion, it is very good intuition, to conclude that set theorists have never thought about this point or have repressed it, because every question either is answered by ridiculous analogies or is closed "as not a real question". You can be sure that many, many people including many "working mathematicians" would like to have an answer. – B ZARKIN

I completely agree. 10 objects cannot be represented by 5 labels. Are you aware to the fact that both numbers are finite ? Countably many things can define uncountably many things (e.g. the rationals define the real numbers), and regardless to that well ordering is not a list of labels. It is a property of a relation. – ASAF KARAGILA

{{And a Cantor list can contain uncountably many entries, because every counter-proof relies on finite numbers for enumeration.}}

You claimed to have studied about Dedekind Cuts, this is a way to define the real numbers from the rationals. Equivalence classes of Cauchy sequences as well. – ASAF KARAGILA

Every Dedekind- and every Cauchy-Sequence is a finite string of symbols, some of them denoting rational numbers. It is impossible, however, to define a real number by an infinite string of rational numbers. To believe so is only a common mistake. – B ZARKIN

For this discussion no source is available because it has been closed and deleted soon as "not a real question".

§ 008 Mathematical logicians often joke that the diagonal method is the only proof method that we have in . This method is the principal idea behind a huge number of fundamental results ... [J D HAMKINS, Nov 22, 2010] http://mathoverflow.net/questions/46970/proofs-of-the-uncountability-of-the-reals/46979

This assertion is correct. And it is a tragedy!

The fundamental mistake of matheology is the conclusion from 1) A finite expression (or number) defines an infinite sequence. to 2) An infinite sequence defines a finite expression (or number). This reversal of implication is of some naivity. But the defenders of that mistake call themselves logicians and admire each other as very intelligent persons.

Nevertheless, even disregarding this error does not help, because the complete infinite Binary Tree can be constructed by countably many infinite paths. That means uncountably many infinite paths cannot be distinguished. The proof is very simple. It is based on the impossibility to find out which paths have been used for construction.

And the notion of uncountability fails for a third reason: Every Cantor-List is capable of providing one or more diagonal numbers. Preconditions for the creation of a diagonal number b are only: 1) The list is well-defined, i.e., all entries ak and all digits akj are known. 2) The replacement rule akk Ø bk is well defined too. Now, it is possible, in principle, to enumerate all Cantor-lists starting from the first one constructed around 1890 to the last one, probably being constructed about 2020. (But even many more could be enumerated because every elementary cell of a temporally and spatially infinite universe belongs to a countable set.) That proves: All diagonals of actually constructed or possible Cantor-lists belong to a countable set. Why can't they be enumerated? They simply don't exist yet!

Would all real numbers be existing and addressable already, then each one could turn up in a Cantor-list, as an infinite bit sequence, at the first place or the second or any other one. Then all these lists could be combined, and diagonalization would contradict the presupposed existence. Hence it is not countability that is disproved by the diagonal argument but simply existence and as such the prerequisite of the notion of complete set as is necessary for set theory.

§ 009 Open letter to a non-mathematician

Georg Cantor's of finished and a time after never, have infected many mathematicians. Their love of these ideas is so strong that they try to punish everyone who dares to present counter arguments. The following case occurred in mathematics stackexchange.com, http://math.stackexchange.com/questions a forum for discussing simple mathematical questions including homework and the like. Usually even contributions of low enjoy a long lifespan. Not so, however, if someone tries to convince freshmen of the irrationality of Cantor's ideas. Moderators and "experienced users" mercilessly delete any heresy. A recent example is this:

Question When Dag Duck receives a heap of dollar coins he counts them all in order to make sure it's the correct amount. In this is often impossible. How much has to be counted in order to be fairly sure about the result? If you get a string of but can't wait until the end-signal is given, how sure can you be to have received the correct meaning?

My answer There is no chance to determine the result unless you have counted the last coin. Similarly, the information is not transferred until transfer has been completed by the end-signal. Otherwise a negation could appear in the subsequent part of the string. All the more it is astonishing that mathematicians are satisfied with Cantor's "enumeration" of the rational numbers. If you count an infinite set like – you never count a share of more than -n limnض 2 = 0 because every is followed by infinitely many others. How can you obtain anything sensible from the result? ("All natural numbers" means all elements of a set of which you cannot have all elements. Infinite means never, not "arrived at after all".) Same is true for infinite "words". They cannot be used for communicating information. Therefore the following reversion of implication, usually applied in set theory, is wrong: A finite formula defines an infinite sequence. ñ An infinite sequence defines a finite formula. We can never obtain a finite formula like 1/9 or ◊2 from an infinite sequence unless we know the last term (which is impossible by definition). An infinite sequence (like Cantor's diagonal) never defines a number. Up to every digit it defines an interval out of countably many. In order to know the limit, you need information about the infinitely many digits to follow. That requires a finite formula.

There is nothing in this answer that could be accused to be wrong or difficult to understand. Nevertheless the answer was deleted after a short time of existence by Henning Makholm who qualified it as "an anti-Cantorian rant with no connection to the question at all" (he was in error, as I know) and Asaf Karagila. The vote of the latter is understandable, because he is preparing his MSc-work in set theory. If it turns out that set theory does not belong to science (as, several decades before, it happened to ), his efforts would have been in vain.

Meanwhile I have been completely excommunicated from math.stackexchange (as before from mathoverflow and some other centers of what I therefore call matheology). In fact, I had expected, and a little bit provoked, the outcome. It proves with breathtaking evidence what everybody not yet infected by Cantor's ideas should know: Many mathematicians are addicts of finished infinity. They refuse to listen to simple , like those discussed in my answer above, and are trying everything to prevent to be cured.

They are not dangerous. Cantor's ideas have not the slightest chance of any application outside of mathematics (because they are blatant humbug) - and even inside they merely cause confusion, namely if is actually taken literally, cp. numbers 1035 to 1044 of http://www.hs-augsburg.de/~mueckenh/KB/KB%201001-.pdf

So be prepared: Many mathematicians believe in finished infinity and may get very angry if you doubt that belief.

§ 010 The present paragraph proves that matheologicans prefer to accept mistakes if otherwise the about their nonsense cannot be suppressed.

In Mathematics StackExchange Seamus had asked the question: Is there a known well ordering of the reals? http://math.stackexchange.com/questions/6501/is-there-a-known-well-ordering-of-the-reals So, from what I understand, the axiom of choice is equivalent to the claim that every set can be well ordered. A set is well ordered by a relation, R if every subset has a least . My question is: has anyone constructed a well ordering on the reals? First, I was going to ask this question about the rationals, but then I realised that if you pick your favourite bijection between rationals and the , this determines a well ordering on the rationals through the natural well order on Ÿ. So it's not the denseness of the reals that makes it hard to well order them. So is it just the size of — that makes it difficult to find a well order for it? Why should that be? - SEAMUS

To be or not to be ... There was much blathering about the existence of a well-ordering of the real numbers.

Goedel explicitly constructed a subset of the reals and a well order on the subset such that (in ZF) it is consistent that the subset is all reals. But subsequently Cohen showed it is also consistent that the subset is not all reals. – G EDGAR

I assume you know the general theorem that, using the axiom of choice, every set can be well ordered. Given that, I think you're asking how hard it is to actually define the well ordering. This is a natural question but it turns out that the answer may be unsatisfying. ... However, there is not even a formula that unequivocally defines a well ordering of the reals in ZFC. ... Worse, it's even consistent with ZFC that no formula in the language of set theory defines a well ordering of the reals (even though one exists). That is, there is a model of ZFC in which no formula defines a well ordering of the reals. – C MUMMERT

No, it's not just the size. ... On the other hand given AC one can obviously write down a well- ordering in a non-constructive way (choose the first element, then the second element, then...). – Q YUAN

First a small error: Ÿ in its natural order is not well-ordered, because Ÿ and many of its subsets do not have a least element. But of course Ù is. And all rationals can be put in bijection with Ù. Second: It is just the size which prevents the well-ordering of —, because a sensible mathematician would refuse to well-order what cannot be distinguished. Take the first, then the second, then the third, ... will not lead further than the countably many possible names reach. You can be sure that never anybody will be able to give a well-ordering of the reals (except those countably many which can be identified). And here "never" means never. Regards, WM

Not necessary to mention that my answer did not live for long. The obvious mistakes ("the natural well order on Ÿ" and "given AC one can obviously write down a well-ordering in a non- constructive way (choose the first element, then the second element, then...)" however, remain there until this very day.

§ 011 God knows a list of all natural numbers [*]. In which way does he memorize the real numbers?

[*] As an example I refer to the totality, the incarnation of all finite positive numbers; this set is a thing by itself and forms, apart from the natural sequence of the involved numbers, a fixed and in all parts defined quantum, an aphorismenon, which obviously must be called larger than every finite number. ... Compare St. Augustin's concurring of the integer sequence as an actually infinite quantum (De civitate Dei. lib. XII, cap. 19): Against those who say God could not know infinite things.

Als Beispiel führe ich die Gesamtheit, den Inbegriff aller endlichen ganzen positiven Zahlen an; diese Menge ist ein Ding für sich und bildet, ganz abgesehen von der natürlichen Folge der dazu gehörigen Zahlen, ein in allen Teilen festes, bestimmtes Quantum, ein aphorismenon, das offenbar größer zu nennen ist als jede endliche Anzahl. [...] Man vgl. die hiermit übereinstimmende Auffassung der ganzen Zahlenreihe als aktual-unendliches Quantum bei S. Augustin (De civitate Dei. lib. XII, cap. 19): Contra eos, qui dicunt ea, quae infinita sunt, nec Dei posse scientia comprehendi. http://agios.org.ua/la/index.php/Aurelius_Augustinus._De_Civitate_Dei_Contra_Paganos._Pars_ 1#LIBER_XII [ (Hrsg.): ", Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Mit erläuternden Anmerkungen sowie mit Ergänzungen aus dem Briefwechsel Cantor - Dedekind. Nebst einem Lebenslauf Cantors von Adolf Fraenkel." Georg Olms Verlagsbuchhandlung, Hildesheim (1966) p. 401] http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN237853094

§ 012 For several terms at Cambridge in 1939, lectured on the philosophical foundations of mathematics. A lecture taught by Wittgenstein, however, hardly resembled a lecture. He sat on a chair in the middle of the room, with some of the class sitting in chairs, some on the floor. He never used notes. He paused frequently, sometimes for several minutes, while he puzzled out a problem. He often asked his listeners questions and reacted to their replies. Many meetings were largely conversation. These lectures were attended by, among others, D. A. T. Gasking, J. N. Findlay, Stephen Toulmin, Alan Turing. [Cora Diamond (ed.): "Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge 1939 from the notes taken by R. G. Bosanquet, Norman Malcolm, Rush Rhees, and Yorick Smythies", The University of Chicago Press, Chicago (1975)] http://www.amazon.com/Wittgensteins-Lectures-Foundations-Mathematics- Cambridge/dp/0226904261/ref=sr_1_1?ie=UTF8&s=books&qid=1248181058&sr=8-1#reader

Imagine set theory's having been invented by a satirist as a kind of parody on mathematics. – Later a reasonable meaning was seen in it and it was incorporated into mathematics. (For if one person can see it as a paradise of mathematicians, why should not another see it as a joke?)

If it were said: "Consideration of the diagonal procedure shews you that the 'real number' has much less analogy with the concept '' than we, being misled by certain analogies, inclined to believe", that would have a good and honest sense. But just the opposite happens: one pretends to compare the "set" of real numbers in magnitude with that of cardinal numbers. The difference in kind between the two conceptions is represented, by a skew form of expression, as difference of extension. I believe, and I hope, that a future generation will laugh at this hocus pocus.

The curse of the invasion of mathematics by is that now any can be represented in a mathematical symbolism, and this makes us feel obliged to understand it. Although of course this method of writing is nothing but the translation of vague ordinary prose.

"Mathematical logic" has completely deformed the thinking of mathematicians and of , by setting up a superficial interpretation of the forms of our everyday language as an analysis of the structures of facts. Of course in this it has only continued to build on the Aristotelian logic.

[Rhees, von Wright, Anscombe (eds.): "Ludwig Wittgenstein, Remarks on the Foundations of Mathematics", -Blackwell (1991)] http://www.amazon.com/Remarks-Foundation-Mathematics-Ludwig- Wittgenstein/dp/0631125051/ref=sr_1_1?ie=UTF8&s=books&qid=1248181778&sr=1-1

§ 013 USER asked in MathOverflow: Did Zermelo and Goedel know about countability of labels? Cantor did not believe that all pieces of information, i.e., all words that can be used for information transfer and all labels that can be used to distinguish objects of thought, belong to a countable set. Is there any evidence that Zermelo or Goedel were aware of that fact?

These kinds of posters are known, in other forums, as "trolls". And, as the saying goes, "Don't feed the trolls". Best to close their questions down and not give them the attention they crave. – L MOSHER {{People who rise above a certain level of fanatism or fall below a certain level of think (as far as they can) that everything they don't understand is trolling.}}

The OP asks "Is there any evidence that Zermelo or Goedel were aware of that fact?" I would ask rather: Is there any evidence for that fact? Specifically, could you please cite a passage in Cantor's writings where he writes about "pieces of information" and "information transfer" and "labels"? – A BLASS

You should know that Cantor spoke and wrote German. Therefore your question is nothing but polemics. And since the present question has been closed, this discussion should cease also. But if you are really interested in objective historical facts of mathematics and not in stirring up hatred, then you may bounce your fellows into translating and not deletíng the question that I will pose immediately. – USER

Letter from Cantor to Hilbert. What did Cantor want to say? On August 8, 1906 Cantor wrote in a letter to Hilbert: "Unendliche Definitionen" (die nicht in endlicher Zeit verlaufen) sind Undinge. Wäre Königs Satz, daß alle „endlich definirbaren" reellen Zahlen einen Inbegriff von der Mächtigkeit ¡0 ausmachen, richtig, so hieße dies, das ganze Zahlencontinuum sei abzählbar, was doch sicherlich falsch ist. Es fragt sich nun, welcher Irrthum liegt dem angeblichen Beweise seines falschen Satzes zu Grunde? Der Irrthum (welcher sich auch in der Note eines Herrn Richard im letzten Hefte der Acta mathematica findet, welche Note Herr Poincaré in dem letzten Hefte der Revue de Métaphysique et de Morale mit Emphase herausstreicht) ist, wie mir scheint, dieser: Es wird vorausgesetzt, dass das System {B} der Begriffe B, welche eventuell zur Definition von reellen Zahlindividuen herangezogen werden müssen, ein endliches oder höchstens abzählbar unendliches sei. Diese Voraussetzung muß ein Irrthum sein, da sich sonst der falsche Satz ergeben würde: "das Zahlencontinuum hat die Mächtigkeit ¡0." What did he mean with this paragraph? - USER

This may be an early attempt to come to grips with the issue of definability, which was still vague at that time. Formal languages and their had not been introduced, and informal languages lead to paradoxes (like the first natural number not definable in fewer than 100 English words) if one takes definability in such languages seriously. Cantor seems convinced that every real number must be definable in some sense, but the sense is not clear. In the first sentence, he dismisses the idea that a definition could itself be an infinite object. He concludes that an uncountable number of must be involved in definitions of real numbers, in order to provide each real number with a finite definition. This contradicts not only our intuitive idea of definability (in a finite or countable language) but also the apparently similar ideas of König, Richard, and Poincaré that Cantor cites. (Note that "herausstreicht" can mean "emphasizes" or "crosses out"; from the context, I infer the former meaning, since Poincaré was not in a position to delete anything from Richard's paper in a different journal.) It would be interesting to see what else Cantor had to say about these uncountably many concepts; presumably he would find it difficult to communicate them. (Does it perhaps lead to ?) - A BLASS

I cannot yet accept your answer because you say: "Cantor seems convinced that every real number must be definable in some sense, but the sense is not clear." This is incorrect. The sense is crystal clear. It must be possible, in finite time (and obviously using a limited bit- frequency), to communicate the complete knowledge concerning an object of mathematics. Example: I say ◊2 and every reader of MO immediately knows what I mean (and can, to any desired digit, find the decimal expansion - but that is not important). – USER

The Cantor quote that you posted is indeed somewhat surprising to me, since he seems to presuppose that all real numbers must be definable in some sense. ... Perhaps the quote shouldn't have been surprising, since clear notions of formal languages and definability arose only later. – A BLASS

Mathematics concerns thinking and communication and is impossible without that. Items that cannot be thought or communicated in finite time do not belong to mathematics. Therefore modern "languages" and the like cannot compensate for the fact that there are not more than countably many words or labels that can be used in mathematics to identify objects. And believing in an order of possibly divine objects that cannot be identified by any finite means is in fact a religious attempt with no relation to math. That's why some call this branch of transfinite set theory "matheology". – USER

Cantor was wrong (we would say) but König used informal reasoning. It may not be well defined in our language which expressions define a real. Also, perhaps the well ordering can not be completely described in the chosen language. - A MEYEROWITZ

"Cantor was wrong", we would not say. Mathematics concerns thinking and communication and is impossible without that. Items that cannot be thought or communicated in finite time do not belong to mathematics. I can assure you: Young, intelligent students having not yet been drilled in transfinity, understand with not a single exception that well-ordering of objects that cannot be identified is an absolutely absurd idea. Ask anybody not yet infected with transfinity. But from the scores of your and Andreas' answer compared to that of my question I can conclude - O, too little space. – USER

It may not be well defined in our language which expressions define a real. But it is very easy to prove that there are not more than countably many expressions. And people loving indirect proofs like Zermelo's well-ordering of the reals should not hesitate to accept also that one as a fact. – USER

There are also less than 36 100 positive integers definable using under 100 letters and numbers. – A MEYEROWITZ

I know that it is hard to define what definability is. Therefore I don't even try. But it is easy to prove what is undefinable, because: "In mathematics and object are equivalent." (Ludwig Wittgenstein) – USER

So instead of using mathematical ideas you draw from natural language which, as discussed between Cantor and Hilbert, is informal and can be interpreted in several ways. Then you use this informality to claim that set theory is ill-defined. I think that your internal logic is somewhat offset. – A KARAGILA

On the contrary. Just the formal definition leads to countability. Unfortunately I have only a German text again: Definiert man die reellen Zahlen in einem streng formalen System, in dem nur endliche Herleitungen und festgelegte Grundzeichen zugelassen werden, so lassen sich diese reellen Zahlen gewiß abzählen, weil ja die Formeln und die Herleitungen auf Grund ihrer konstruktiven Erklärungen abzählbar sind. [Kurt Schütte: "Beweistheorie", Springer (1960)] Look up in your library: K. Schütte, proof theory. (Schütte was a pupil of Hilbert.) – USER

§ 014 On Jan. 31, 2012 I answered in MatheOverflow a question concerning the infinite Binary Tree: The set of all finite paths (from the root-node to any other node) in the complete infinite Binary Tree is countable. Therefore the complete infinite Binary Tree has countably many paths that can be identified by nodes. It is impossible to identify an infinite path by nodes, because 1) every node belongs to a finite path, and 2) there is no identification unless it has been finished. Therefore an infinite path can only be identified by a finite expression like "always turn left", or "0.111...", or "the path which represents 1/π", or simply "1/3". However, the set of finite expressions has countable cardinality. Therefore the set of all paths in the complete infinite Binary Tree has countable cardinality. Regards, WM

For a real number r > 1 wouldn´t you consider "the path which represents 1/r" to be a path? Aren´t there uncountably many real numbers r > 1? – R DE LA VEGA

For a real number r > 1, that can be defined by a finite expression, 1/r represents a path. But there are not more than countably many finite expressions. – WM

Further questions were not admitted. My answer had been deleted.

§ 015 Belsa Zarkin asked in Mathematics.StackExchange: All rational numbers of the unit interval [0, 1] can be covered by countably many intervals, such n that the n-th rational is covered by an interval In of measure 1/10 . There remain countably many complementary intervals of measure 8/9 in total. Does each of the complementary intervals contain only one ? Then there would be only countably many which could be covered by another set of countably many intervals of measure 1/9. Is there at least one of the complementary intervals containing more than one irrational number? Then there are at least two irrational numbers without a rational between them. That is mathematically impossible.

There is no interval left behind. Every interval contains at least one rational number, so if you remove all rational numbers (let alone a bunch of intervals containing all of them), there can be no interval left over. – BRUNO {{Intervall, Teilmenge I einer totalen Ordnung (M, ≤). Insbesondere sind also die leere Menge « und M selbst Intervalle. ["Lexikon der Mathematik", Spektrum (2003)]}}

There is a mistake in your argument, which follows from the fact that the complement of this union contains no interval. ...The irrational numbers form a totally disconnected space, namely every connected component is a singleton.- A KARAGILA

In a totally disconnected space there must be points or intervals disconnecting it. The number of these points or intervals is countable and in bijection with the remaining intervals. The bijection is the same as that from Ù to –. (For instance at every step n the configuration of intervals could be determined in principle.) If you deny the of this bijection for the limit, why don't you deny the validity for the limit of the bijection from Ù to –? – B ZARKIN

The definition of a totally disconnected space is a space in which every connected space is a singleton. E.g. a discrete space. Are you saying that every discrete space is countable? – A KARAGILA

This discussion lasted for a while. Meanwhile it has been deleted. Correct mathematics is this: -n There are countably many intervals In of measure 10 such that In covers the rational qn. Then 1/9 (or less) of the unit interval is covered. In the remaining 8/9 (or more) there are uncountably many irrationals. But every two irrationals have a rational between each other. That implies two irrationals have at least one interval In between each other (because there are no rationals outside of intervals I ). That implies two irrationals have at least one of ¡ endpoints I or I of n 0 n1 n2 intervals I between each other. These endpoints can be considered structuring and n1 enumerating a Cantor-list. The only difference is that the enumeration does not follow the natural order of Ù. But the number of naturals does not change by reordering. So we have uncountably many irrationals separated by countably many endpoints. That is a contradiction similar to uncountably many entries in a Cantor-list or uncountably many terms in a sequence.

§ 016 If we define the real numbers in a strictly formal system, where only finite derivations and fixed symbols are permitted, then these real numbers can certainly be enumerated because the formulas and derivations on the of their constructive definition are countable.

Definiert man die reellen Zahlen in einem streng formalen System, in dem nur endliche Herleitungen und festgelegte Grundzeichen zugelassen werden, so lassen sich diese reellen Zahlen gewiß abzählen, weil ja die Formeln und die Herleitungen auf Grund ihrer konstruktiven Erklärungen abzählbar sind. [Kurt Schütte: "Beweistheorie", Springer (1960)] http://www.amazon.de/Beweistheorie-Kurt- Sch%C3%BCtte/dp/B0000BNKI7/ref=sr_1_1?s=books&ie=UTF8&qid=1286292242&sr=1-1

§ 017 Thesis XIII in Brouwers Dissertation (Appendix): "Over de grondslagen der wiskunde" (Februari 1907, Dutch) simply reads: "De tweede getalklasse van Cantor bestaat niet", translated: Cantors second number class does not exist. {{That is an acceptable foundation of acceptable mathematics.}} http://www.archive.org/details/overdegrondslag00brougoog

He quickly discovered that his ideas on the foundations of mathematics would not be readily accepted. {{His ideas would devastate matheology.}} http://www-history.mcs.st-andrews.ac.uk/Biographies/Brouwer.html

§ 018 Feferman and Levy showed that one cannot prove that there is any non-denumerable set of real numbers which can be well ordered. Moreover, they also showed that the that the set of all real numbers is the union of a denumerable set of denumerable sets cannot be refuted. [Abraham A. Fraenkel, Yehoshua Bar-Hillel, Azriel Levy: "Foundations of Set Theory", North Holland, Amsterdam (1973) p. 62] http://www.amazon.de/gp/product/0720422701/ref=sib_rdr_dp

§ 019 Cantor's 'paradise' as well as all modern axiomatic set theory [AST] is based on the (self- contradictory) concept of actual infinity. Cantor emphasized plainly and constantly that all transfinite objects of his set theory are based on the actual infinity. Modern AST-people try to persuade us to believe that the AST does not use actual infinity. {{Many don't even know the difference.}} It is an intentional and blatant lie, since if infinite sets, ◊ and Ù, are potential, then the uncountability of the continuum becomes unprovable, but without the notorious uncountablity of continuum the modern AST as a whole transforms into a long twaddle about nothing ... [Letter from A. A. Zenkin to D. Zeilberger] http://www.math.rutgers.edu/~zeilberg/fb68.html

Prof. Dr. Alexander A. Zenkin (1937 - 2006) was leading research scientist of the computer center of the Russian Academy of Sciences.

§ 020 Epistola Pentecostes MDCCCLXXXVIII

I have no doubt concerning the truth of the transfinitum, which I have recognized with the help of God. [...] I happen to be a somewhat familiar not only with mathematics but also with several other sciences. Therefore I am able to compare theorems, here and there, with respect to their objective certainty. From no other subjects of the created I have a safer and, if this expression is allowed, a more certain realization than of the theorems of - and type-theories. That's why I am convinced that this theory one day will belong to the common property of objective science and will be confirmed in particular by that theology which is based upon the holy bible, tradition and the natural disposition of the human race - these three necessarily being in harmony with each other.

If one chooses this foundation for the doctrine of actual infinity, one stands firm and is, I might almost say, easily able to reject all the objections which have been devised over millenia against the infinite numbers, and to reduce them to their apparent .

I completely agree that [...] all the finite (and, to a much higher degree, all the transfinite), from a diversity of aspects, points to the Absolute, i.e., the existence of the Absolute can necessarily be proved by a dialectical rational conclusion, in accordance with Bonaventura's sentence: Invariable rules (of human reason) are rooted in the eternal light and lead to it.

"Couldn't God, after having created an infinite set of stones or angels, create further angels?" {{asks Durandus de Sancto Porciano, OP.}} Of course he can do that, must be answered. When he then continues to conclude: "Therefore the angels created at first were not infinitely many." so is this conclusion utterly wrong, because the supposed set of created angels is a transfinitum that can be increased as well as decreased.

[Georg Cantor, Letter of Pentecost 1888 to P. Ignatius Jeiler, OFM {{that does not mean "Online Football Manager" but "Ordo Fratrum Minorum", order created by Francis of Assisi}}, Praefect. Coll. S. Bonav., quoted in C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit." Vol. 53, Franz Steiner Verlag (2005) p. 410ff] http://www.steiner-verlag.de/programm/fachbuch/geschichte/universitaets-und- wissenschaftsgeschichte/reihen/view/titel/54670.html

§ 021 Epistola Pentecostes MDCCCLXXXVIII

Understanding of the theory of the transfinite does not require scholarly preparation in newer mathematics {{modern matheologians hold another opinion: who does not believe in the diagonal proof cannot have understood it and lacks mathematical skills}}; it could be rather damaging than helpful, because modern mathematics [...] has been perverted to materialistic one-sidedness and has been blinded for any objective metaphysical recognition and, therefore, for its own foundations too. Whereas the theory of the transfinite for its foundations does not need the so called analysis (), the latter can neither lay its own groundwork nor proceed to completing its edifice without the former. Under the first aspect I mention the theory of irrational numbers which need the transfinite for a proper foundation, further there is the advanced theory of functions that already today raises questions which without the aid of the theory of transfinite order types and cardinal numbers cannot even be articulated, let alone be solved. Yet it is possible for everyone, in particular for an educated , to scrutinize the of transfinite number theory and to become convinced of its correctness {{and those who do not?}}.

You emphasize with full right, Reverend Father, that according to nearly all teachings of the old school (for me the only authoritative one) the divine intellect recognizes with respect to "objecta extra ipsum cognita [...] infinita actu et categorematice“, not always "a parte rei" but always in divine recognition "simultatem habent in esse cognito". If only this quite safe and unshakeable sentence had always been realized in its full contents (i.e., not only in general but also in its special meaning, in concreto) then one would have recognized without much effort the truth of the transfinite, and quarrel and errors of thousands of years would have been avoided.

Applying this sentence to a special class of objects of the divine recognition, we arrive at the elements of transfinite theory of numbers and types. Every single finite cardinal number (1 or 2 or 3 etc.) is contained in the divine intellect in form of an exemplary idea as well as a unified form for the recognition of uncountably many composed things that belong to that very cardinal number. Hence, all finite cardinal numbers are separately and simultaneously present in God's . (Cp S. Augustin, De civitate Dei, lib. XII, cap. 19: contra eos, qui dicunt ea, quae inf. sunt, nec Dei scientia comprehendi) ("Against those who state that the infinite could not be comprehended by the knowledge of God")

In their totality they form a diversified unity, a thing, separated from the remaining contents of the divine intellect, that in itself again is a subject of divine recognition. But since recognition of a thing requires a uniform shape by which the thing can be recognized, there must be in God's mind a certain cardinal number that in the same way is related to the set or totality of all finite cardinal numbers as, for instance, the cardinal number 7 is related to the set of tones of the scale [C, D, E, F, G, A, H] of an octave.

[Georg Cantor, Letter of Pentecost 1888 to P. Ignatius Jeiler, OFM, quoted in C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 414ff] http://www.steiner-verlag.de/programm/fachbuch/geschichte/universitaets-und- wissenschaftsgeschichte/reihen/view/titel/54670.html

§ 022 All rational numbers of the unit interval [0, 1] can be covered by countably many intervals, such that the n-th rational is covered by an interval of measure 1/10n. There remain countably many complementary intervals of measure 8/9 in total. Does each of the complementary intervals contain only one irrational number? Then there would be only countably many which could be covered by another set of countably many intervals of measure 1/9. My question: Can this contradiction be formalized in ZFC? [user31686, April 15, 2012] http://math.stackexchange.com/questions/132022/formalizing-an-idea/150674#150674

It has been mentioned already that the irrationals ξα of the set Ξ of the remaining part of measure 8/9 (or more), that is not covered by your intervals, form a totally disconnected space, so called "Cantor dust". Every particle ξα œ Ξ is separated from every other particle ξβ œ Ξ by at least one rational qn, and, as every qn is covered by an interval In, it is separated by at least one interval In. Since the end points an and bn of the In are rational numbers too, also being covered by their own intervals, the particles of Cantor dust can only be limits of infinite sequences (an) or (bn) of endpoints of overlapping intervals In. (If they don't overlap, then the limits must come earlier, but in any case infinitely many endpoints are required to form a limit.) Such an infinite set of overlapping intervals is called a cluster. In principle, given a fixed enumeration of the rationals, we can calculate every cluster Ck and the limits of its union. Since two clusters are disjoint (by their limits), there are only countably many clusters (disjoint subsets of the countable set of intervals In). Therefore, every irrational ξα can be put in bijection with the cluster lying right of it, say, between ξα and its next right neighbour ξβ. (Note that there is no next irrational to ξα but there is a next right ξβ œ Ξ to ξα.) So, by this bijection we prove that the set of uncovered irrational numbers ξα œ Ξ is countable. [Stentor Schicklgruber, StackExchange (2012)] http://math.stackexchange.com/users/32353/stentor-schicklgruber

Let all rational numbers qn of the interval (0,¶) be covered by intervals In = [sn, tn] of measure −n |In| = 2 , such that qn is the center of In. Then there remain uncountably many irrational numbers as uncovered "Cantor dust". Every uncovered irrational xα must be separated from every uncovered irrational xβ by at least one rational, hence by at least one interval In covering that rational. But as the end points sn and tn of the In also are rational numbers and also are covered by their own intervals, the irrationals xα can only be limits of infinite sequences (sn) or (tn) of endpoints of overlapping intervals In. In principle we can calculate the limit xα of every such sequence (sn) or (tn) of endpoints of overlapping intervals. Therefore, every irrational xα can be put in bijection with the infinite set of intervals lying right of it, say, between xα and its right neighbour xβ. There are countably many disjoint sets like {t | t œ (tn)} of elements of the sequences (tn) converging to one of the xα. By this bijection we get a countable set of not covered irrational numbers xα. Where are the other irrational numbers that are not covered by intervals In? Nowhere. Uncountability is contradicted. [Quidquid pro quo, MathOverflow (2012)] http://mathoverflow.net/users/24011/quidquid-pro-quo

§ 023 The true reason for the incompleteness that is inherent in all formal systems of mathematics lies in the fact that the generation of higher and higher types can be continued into the transfinite whereas every formal system contains at most countably many. This will be shown in part II of this paper. {{Part II was never published.}} In fact we can show that the undecidable statements presented here always become decidable by adjunction of suitable higher types (e.g., adding the type ω to system P). Same holds for the axiom system of set theory.

Der wahre Grund für die Unvollständigkeit, welche allen formalen Systemen der Mathematik anhaftet, liegt, wie im II. Teil dieser Abhandlung gezeigt werden wird, darin, daß die Bildung immer höherer Typen sich ins Transfinite fortsetzen läßt [...] während in jedem formalen System höchstens abzählbar viele vorhanden sind. Man kann nämlich zeigen, daß die hier aufgestellten unentscheidbaren Sätze durch Adjunktion passender höherer Typen (z. B. des Typus ω zum System P) immer entscheidbar werden. Analoges gilt auch für das Axiomensystem der Mengenlehre. [Kurt Gödel: "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I", Monatshefte für Mathematik und Physik 38 (1931) p. 191]

§ 024 Simultaneously I send, as printed matter, a curious booklet by , which I let you have, since I happen to possess another one. Although there is much, perhaps the most, in error, it was very stimulating for me, in particular because of the opposition which it has aroused in me. [Cantor to Dedekind, Oct. 7, 1882]

I never fully agreed with Dedekind's paper. [Cantor to Jourdain, July 18, 1901]

In vol. XXII, no. II of the Annalen pag. 249, Klein has his paper about function strips printed again which I always have considered the non plus ultra of higher nonsense, although Mr. Klein refers to the wisdom of Mr. Kronecker. If you have read my essay Grundlagen attentively, you will find, that I, on pag. 9, 10, 11 and on pag. 19 and 20, attack and condemn just these opinions of Kronecker in strongest terms; on pag. 20, in another matter, I make him a compliment, together with Dedekind however (that will very much annoy him). [Cantor to Mittag-Leffler, Sept. 9, 1883]

The first old auntie said, we may Now try to fix it swift. Our little Sophy, at Saints day, Shall get from us a gift.

Of course, the second aunt said keen, I think we should acquire A summer dress in bright pea green. That's not at all her desire.

The third old aunt agreed and said: With yellow stripes, old style. I know she'll be annoyed and yet She'll have to thank us and smile.

[Wilhelm Busch: "Die erste alte Tante sprach", Kritik des Herzens (1874)] (Well, more or less.) The German original can be found here: http://de.wikisource.org/wiki/Die_erste_alte_Tante_sprach In the same year Cantor published his first paper on set theory: About a property of the set of all real algebraic numbers [Crelles Journal f. Mathematik, vol 77, 258-262 (1874)].

Many thanks for your information about the French translation of Paul du Bois Reymond's miserable paper. [Cantor to Mittag-Leffler, Aug. 4, 1888]

And the "Acta" shall be abused to disseminate this dirty rubbish? He will not use his own journal. [...] for my own works I demand bias, but not for my transitory person, but bias for the truth which is eternal and with superior contempt looks down upon the subversives (among others Kronecker), who dare to imagine, they could lastingly change it with their dreadful scribblings. [Cantor to Mittag-Leffler, Jan. 26, 1884]

I am quite an adversary of Old Kant, who, in my eyes has done much harm and mischief to philosophy, even to mankind; as you easily see by the most perverted development of in in all that followed him, as in Fichte, Schelling, Hegel, Herbart, Schopenhauer, Hartmann, Nietzsche, etc. etc. on to this very day. I never could understand that and why such reasonable and enabled peoples as the Italiens, the English and the French are, could follow yonder sophistical philistine, who was so bad a mathematician. And now it is that in just this abominable mummy, as Kant is, Monsieur Poincaré felt quite enamoured, if he is not bewitched by him. [Cantor to Russell, Sept. 19, 1911 (English by Cantor)]

With mathematicians, no cheerful relationship can be obtained. [Goethe to Zelter, Jan. 18, 1823] Originally Goethe wrote (but that does not fit so well): "With philologists and mathematicians, no cheerful relationship can be obtained.".

For original German texts see "Das Kalenderblatt 090727": http://www.hs-augsburg.de/~mueckenh/KB/KB%20001-200.pdf

§ 025 Modern mathematics as religion [...] Most (but not all) of the difficulties of Set Theory arise from the insistence that there exist 'infinite sets', and that it is the job of mathematics to study them and use them. In perpetuating these notions, modern mathematics takes on many of the aspects of a religion. It has its essential creed - namely Set Theory, and its unquestioned assumptions, namely that mathematics is based on 'Axioms', in particular the Zermelo-Fraenkel 'Axioms of Set Theory'. It has its anointed priesthood, the logicians, who specialize in studying the foundations of mathematics, a supposedly deep and difficult subject that requires years of devotion to master. Other mathematicians learn to invoke the official mantras when questioned by outsiders, but have only a hazy view about how the elementary aspects of the subject hang together logically. Training of the young is like that in secret societies - immersion in the cult involves intensive undergraduate memorization of the standard thoughts before they are properly understood, so that comprehension often follows belief instead of the other (more healthy) way around. A long and often painful graduate school apprenticeship keeps the cadet busy jumping through the many required hoops, discourages critical thought about the foundations of the subject, but then gradually yields to the gentle acceptance and support of the brotherhood. The ever-present demons of inadequacy, failure and banishment are however never far from view, ensuring that most stay on the well-trodden path. The large international conferences let the fellowship gather together and congratulate themselves on the uniformity and sanity of their world view, though to the rare outsider that sneaks into such events the proceedings no doubt seem characterized by jargon, mutual incomprehensibility and irrelevance to the outside world. The official doctrine is that all views and opinions are valued if they contain truth, and that ultimately only elegance and utility decide what gets studied. The reality is less ennobling - the usual hierarchical structures reward allegiance, conformity and technical mastery of the doctrines, elevate the interests of the powerful, and discourage dissent. There is no evil intent or ugly conspiracy here - the practice is held in place by a mixture of well-meaning effort, inertia and self-interest. We humans have a fondness for believing what those around us do, and a willingness to mold our intellectual constructs to support those hypotheses which justify our habits and make us feel good. [N J Wildberger: "Set Theory: Should You Believe?"] http://web.maths.unsw.edu.au/~norman/views2.htm

§ 026 {{Good mathematicians are precise and clear. Nevertheless, leading exegetes of matheology do not seldom find the correct meaning, often the contrary of the written text, between the lines. The following paragraph shall easen their effort.}}

(i) Infinite totalities do not exist in any proper sense of the word

(i.e., either really or ideally).

More precisely, any mention, or purported mention, of infinite totalities is, literally meaningless.

{{This text is not at all put into any perspective by the following paragraph (here the extra lines will hardly interest the matheologian and, therefore, have been omitted).}}

(ii) Nevertheless, we should continue the business of Mathematics "as usual," i.e., we should act as if infinite totalities really existed. [...] I must regard a theory which refers to an infinite totality as meaningless in the sense that its terms and sentences cannot posses the direct interpretation in an actual structure that we should expect them to have by analogy with concrete (e.g., empirical) situations. This is not to say that such a theory is pointless or devoid of significance. {{Of course this is neither to say the contrary.}}

[A. Robinson: "Formalism 64" in W.A.J. Luxemburg, S. Koerner (eds.): "A. Robinson: Selected Papers", North Holland, Amsterdam (1979)]

§ 027 By hindsight, it is not surprising that there exist undecidable , as meta- proved by Kurt Gödel. Why should they be decidable, being meaningless to begin with! The tiny fraction of first order statements that are decidable are exactly those for which either the statement itself, or its negation, happen to be true for symbolic integers. A priori, every statement that starts "for every integer n" is completely meaningless. [Doron Zeilberger: "'Real' analysis is a degenerate case of discrete analysis"] http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf

§ 028 The expression "and so on" is nothing but the expression "and so on" [...] the sign "1, 1+1, 1+1+1 ..." is to be taken as perfectly exact; governed by definite rules which are different from those for "1, 1+1, 1+1+1", and not a substitute for a series "which cannot be written down".

There is no such thing as "the cardinal numbers", but only "cardinal numbers" and the concept, the form "cardinal number". Now we say "the number of the cardinal numbers is smaller than the number of the real numbers" and we imagine that we could perhaps write the two series side by side (if only we weren't weak humans) and then the one series would end in endlessness, whereas the other would go on beyond it into the actual infinite. But this is all nonsense.

"This proposition is proved for all numbers by the recursive procedure". That is the expression that is so very misleading. It sounds as if here a proposition saying that such and such holds for all cardinal numbers is proved true by a particular route, or as if this route was a route through a space of conceivable routes. But really the recursion shows nothing but itself, just as periodicity too shows nothing but itself".

After all I have already said, it may sound trivial if I now say that the mistake in the set- theoretical approach consists time and again in treating laws and enumerations (lists) as essentially the same kind of thing and arranging them in parallel series so that one fills in gaps left by the other.

[L. Wittgenstein: "Philosophical Grammar", Basil Blackwell, Oxford (1969)]

§ 029 One of the characteristic features of Cantorism is that, instead of rising to the general by erecting more and more complicated constructions, and defining by construction, it starts with the genus supremum and only defines, as the scholastics would have said, per genus proximum et differentiam specificam. Hence the horror he has sometimes inspired in certain , such as Hermitte's, whose favourite idea was to compare the mathematical with the natural sciences. For the greater number of us these prejudices had been dissipated, but it has come about that we have run against certain paradoxes and apparent contradictions, which would have rejoiced the heart of Zeno of Elea and the school of Megara. Then began the business of searching for a remedy, each man his own way. For my part I think, and I am not alone in so thinking, that the important thing is never to introduce any entities but such as can be completely defined in a finite number of words. Whatever be the remedy adopted, we can promise ourselves the joy of the doctor called in to follow a fine pathological case. [pp. 44f] There is no actual infinity. The Cantorians forgot this, and so fell into contradiction. [p. 195] [Henri Poincaré: "Science and Method", Translated by Francis Maitland, Nelson, London (1914)] http://archive.org/stream/sciencemethod00poinuoft#page/194/mode/2up

§ 030 We can create in mathematics nothing but finite sequences, and further, on the ground of the clearly conceived "and so on", the order type omega, but only consisting of equal elements {{i.e. numbers like 0,999...}}, so that we can never imagine the arbitrary infinite binary fractions as finished {{Brouwers Thesis, p. 143}}. [Dirk van Dalen: "Mystic, Geometer, and Intuitionist: The Life of L.E.J. Brouwer", Oxford University Press (2002)]

Certainly we can also create other infinite sequences by finite expressions like 0,101010... that would read 0,AAA... in hexadecimal notation. In that way also every other rational number can be written.

But irrational numbers in fact are not available as sequences of digits.

Proof: Construct the Binary Tree by all finite paths that have an infinite sequence of 000...appended. Construct the Binary Tree by all finite paths that have an infinite sequence of 111... appended. Construct the Binary Tree by all finite paths that have an infinite sequence of 010101... appended. Construct the Binary Tree by all finite paths that have an infinite sequence of all bits of 1/π appended. The reader will not be able to distinguish the constructed Binary Trees and to determine what infinite paths are missing.

This fine proof has the advantage that matheologians cannot dispute it. They simply are not able to distinguish numbers by sequences of bits or digits.

§ 031 Let us distinguish between the genetic, in the dictionary sense of pertaining to origins, and the formal. Numerals (terms containing only the unary function symbol S and the constant 0) are genetic; they are formed by human activity. All of mathematical activity is genetic, though the subject matter is formal. Numerals constitute a potential infinity. Given any numeral, we can construct a new numeral by prefixing it with S. Now imagine this potential infinity to be completed. Imagine the inexhaustible process of constructing numerals somehow to have been finished, and call the result the set of all numbers, denoted by Ù. Thus Ù is thought to be an actual infinity or a completed infinity. This is curious terminology, since the etymology of “infinite” is “not finished”. We were warned. : Infinity is always potential, never actual. Gauss: I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. We ignored the warnings. With the work of Dedekind, Peano, and Cantor above all, completed infinity was accepted into mathematics. Mathematics became a faith-based initiative. Try to imagine Ù as if it were real. A friend of mine came across the following on the Web:

www.completedinfinity.com Buy a copy of Ù! Contains zero - contains the successor of everything it contains - contains only these. Just $100. Do the math! What is the price per number? Satisfaction guaranteed! Use our secure form to enter your credit card number and its security number, zip code, social security number, bank’s routing number, checking account number, date of birth, and mother’s maiden name. The product will be shipped to you within two business days in a plain wrapper.

My friend answered this ad and proudly showed his copy of Ù to me. I noticed that zero was green, and that the successor of every green number was green, but that his model contained a red number. I told my friend that he had been cheated, and had bought a nonstandard model, but he is color blind and could not see my point. I bought a model from another dealer and am quite pleased with it. My friend maintains that it contains an ineffable number, although zero is effable and the successor of every effable number is effable, but I don’t know what he is talking about. I think he is just jealous. The point of this conceit is that it is impossible to characterize Ù unambiguously, as we shall argue in detail. [...] Over two and a half millennia after Pythagoras, most mathematicians continue to hold a religious belief in Ù as an object existing independently of formal human construction.

[Edward Nelson: "Hilbert’s Mistake" (2007)] http://www.math.princeton.edu/~nelson/papers/hm.pdf

§ 032 Although Zermelo-Fraenkel set theory (ZFC) is generally accepted as the appropriate foundation for modern mathematics, proof theorists have known for decades that virtually all mainstream mathematics can actually be formalized in much weaker systems which are essentially number-theoretic in nature. [...] not only is it possible to formalize core mathematics in these weaker systems, they are in important ways better suited to the task than ZFC [...] most if not all of the already rare examples of mainstream theorems whose proofs are currently thought to require metaphysically substantial set-theoretic principles actually do not; and set theory itself, as it is actually practiced, is best understood in formalist, not platonic, terms, so that in a real sense set theory is not even indispensable for set theory. [...] set theory should not be considered central to mathematics. Probably most mathematicians are more willing to be platonists about number theory than about set theory, in the “truth ” sense that they firmly believe every sentence of first order number theory has a definite truth , but are less certain this is the case for set theory. Those mathematicians who are unwilling to affirm that the twin primes conjecture, for example, is objectively true or false are undoubtedly in a small minority; in contrast, suspicion that questions like the continuum hypothesis or the existence of measurable cardinals may have no genuine seems fairly widespread. Some possible reasons for this difference in attitudes towards number theory and set theory are (1) a sense that natural numbers are evident and accessible in a way that arbitrary sets are not; (2) suspicion that sets are philosophically dubious in a way that numbers are not; (3) the existence of truly basic set-theoretic questions such as the continuum hypothesis which are known to be undecidable on the basis of the standard axioms of set theory, and the absence of comparable cases in number theory; and (4) the fact that is inconsistent. The classical paradoxes of naive set theory particularly cast doubt on the idea of a well-defined canonical universe of sets in which all set-theoretic questions have definite answers. One philosophically important way in which numbers and sets, as they are naively understood, differ is that numbers are physically instantiated in a way that sets are not. Five apples are an instance of the number 5 and a pair of shoes is an instance of the number 2, but there is nothing obvious that we can analogously point to as an instance of, say, the set {{«}}. [...] Unfortunately, the philosophical difficulties with set-theoretic objects platonism are extremely severe. First, there is the ontological problem of saying just what sets are.[...] Perhaps the most influential philosophical defense of set theory is the Quine-Putnam indispensability argument. According to this argument, mathematics is indispensable for various established scientific theories, and therefore any evidence that confirms these theories also confirms the received foundation for mathematics, namely set theory. But as a result of work of many people going back to , we now know that the kind of mathematics that is used in scientific applications is not inherently set-theoretic, and indeed can be developed along purely number-theoretic lines. This point has been especially emphasized by Feferman. Consequently, contrary to Quine and Putnam, the confirmation of present-day scientific theories provides no special support for set theory. [...] This raises the possibility that the use of set theory as a foundation for mathematics may be an historical aberration. We may ultimately find that ZFC really has no compelling justification and is completely irrelevant to ordinary mathematical practice. [Nik Weaver: "Is set theory indispensable?"] http://www.math.wustl.edu/~nweaver/indisp.pdf

§ 033 The clear understanding of formalism in mathematics has led to a rather fixed dogmatic position which reads: Mathematics is what can be done within axiomatic set theory using classical predicate logic. I call this doctrine the Grand Set Theoretic Foundation. [...] Even in the 1940's, with the growth of abstract algebra, axiomatic set theory was not regarded as a central doctrine. It was not until about 1950 that the Grand Set Theoretic Foundation was finally complete and officially accepted under the slogan which might have read: "Mathematics is exactly that subject which can be developed by logical rules of proof from the Zermelo-Fraenkel axioms for set theory." This foundation scheme had its popular version in the "new math" for schools. It also had its philosophical doctrine, a version of Platonism, that the world of sets is that constructed in the standard of all ranked sets. [...] The Zermelo- Fraenkel axioms are then (a selection of) the facts true for all sets in this hierarchy. This is sometimes claimed to describe the ultimate Platonic reality which underlies all mathematics: Perhaps the Zermelo-Fraenkel axioms do not describe everything, but with a little more insight we will understand all the axioms necessary and then at least in principle all mathematical problems can be settled from the axioms. It is my contention that this Grand Set Theoretic Foundation is a mistakenly one-sided view of mathematics and also that its precursor doctrine (Dedekind cuts) was also one-sided. [Saunders Mac Lane: "Mathematical models: A sketch for the philosophy of mathematics", The American Mathematical Monthly, Vol. 88, No. 7 (1981) 462-472] http://en.wikipedia.org/wiki/MacLane Don't miss his story: The man who could have changed history! http://www.ams.org/notices/199510/maclane.pdf (p. 1136, first paragraph)

§ 034 Announcing some congresses that may be of highest interest http://ufocongress.com/grant-cameron-added-to-the-2013-iufoc-speakers/ http://www.impan.pl/~set_theory/Conference2012/ http://www.clevelandufo.com/?p=458 http://www.uacastrology.com/schedule2012.shtml http://www.creationicc.org/ to the participants - but not to anybody else.

Therefore it is no disadvantage that some have ceased already. (Further it is claimed that congresses on activities can also be attended in the past - by means of paranormal activities, of course.)

§ 035 Borel's often-expressed credo is that a real number is really real only if it can be expressed, only if it can be uniquely defined, using a finite number of words. It's only real if it can be named or specifed as an individual mathematical object. {{How else could it be? Numbers do not acquire existence other than that lent to them by (human) minds. Minds who mean to be able to lend meanings that they, in principle, cannot, and who, in addition, mean to be able to prove that impossibility as possible, appear to be mean minds - with respect to the spirit of truth.}} And in order to do this we must necessarily employ some particular language, e.g., French. Whatever the choice of language, there will only be a countable infinity of possible texts, since these can be listed in size order, and among texts of the same size, in alphabetical order. This has the devastating consequence that there are only a denumerable infinity of such "accessible" reals, and therefore [...] the set of accessible reals has measure zero. So, in Borel's view, most reals, with one, are mathematical fantasies, because there is no way to specify them uniquely. This is a more refined version of Borel's idea [Borel, 1960] of defining the complexity of a real number to be the number of words required to name it. Why should we believe in real numbers, if most of them are uncomputable? Why should we believe in real numbers, if most of them, it turns out, are maximally unknowable like Ω?

The latest strong hints in the direction of discreteness come from quantum gravity [...], in particular from the Bekenstein bound and the so-called "holographic principle." According to these ideas the amount of information in any physical system is bounded, i.e., is a finite number of 0/1 bits. But it is not just fundamental physics that is pushing us in this direction. Other hints come from our pervasive digital technology, from molecular biology where DNA is the digital software for life, and from a priori philosophical prejudices going back to the ancient Greeks. According to Pythagoras everything is number, and God is a mathematician. This point of view has worked pretty well throughout the development of modern science. However now a neo- Pythagorean doctrine is emerging, according to which everything is 0/1 bits, and the world is built entirely out of digital information. In other words, now everything is software, God is a computer programmer, not a mathematician, and the world is a giant information-processing system, a giant computer.

[Gregory Chaitin: "How real are real numbers?" (2004)] http://arxiv.org/abs/math.HO/0411418

§ 036 An argument against so called cranks (i.e., those who neither do believe in finished infinity nor in undefinable definitions) is the claim that they do not only vehemently deny set theory but as vehemently contradict each other. That may happen. But it is not different in matheology and, therefore, cannot serve as an argument in favour of the superiority of the latter. Here are some examples:

Hitherto logicians have only operated with chains consisting of a finite number of conclusions. But since Cantor mathematicians operate with infinite chains. [A. Schoenflies: "Über die logischen Paradoxien der Mengenlehre", Jahresbericht der Deutschen Mathematiker- Vereinigung, (1906) 19-25.]

When paying attention you can find the mathematical literature flooded with inconsistencies and thoughtlessness that in most cases are caused by the infinite. For instance, if, as a restricting requirement, it is emphasized that in strict mathematics only a finite number of conclusions is admissible in a proof - as if anybody ever had succeeded in drawing infinitely many conclusions! [D. Hilbert: "Über das Unendliche", Math. Annalen 95 (1925) 161-190]

These are Cantor's first transfinite numbers, the numbers of the second number class, as Cantor calls them. We reach them simply by counting across the ordinary infinite. [Hilbert, loc. cit.]

Here Hilbert even has contradicted himself, because counting is nothing else but concluding from n on n + 1. And there is another self-contradiction in one and the same paper:

No one shall drive us from the paradise which Cantor has created for us. [...] his theory of transfinite numbers; it appears to me as the most admirable blossom of mathematical spirit and one of the supreme achievements of purely intellectual human activity. What are the facts of the matter? [...] the infinite is nowhere realized; neither is it present in nature nor is it admissible as a foundation of our cerebral activity - a remarkable harmony between being and thinking. [Hilbert, loc. cit.]

Kant already has taught us - and this is an integral part of his teaching - that mathematical contents is independent of logic; mathematics can never be founded by logic alone.[Hilbert, loc. cit.]

I am quite an adversary of Old Kant, who, in my eyes has done much harm and mischief to philosophy, even to mankind. [G. Cantor to B. Russell, Sept. 19, 1911]

So Kant's achievements are quite controversially judged by Cantor and Hilbert. Frege's and Dedekind's achievements are controversially judged even by Hilbert alone:

Brouwer is not, as Weyl thinks, the revolution, but the attempted repetition of a putsch with old means, that in the past, undertaken much more dashingly, nevertheless completely failed, and now is condemned to fail because the state power is well armed by Frege, Dedekind, and Cantor. [D. Hilbert: "Die Neubegründung der Mathematik. Erste Mitt.", Abhandl. d. Math. Seminars d. Univ. Hamburg, Bd. 1 (1922), S. 157-177]

... the contents of mathematics is independent of logic and therefore never ever can be founded by logic alone. That's why the efforts of Frege and Dedekind were condemned to fail. [D. Hilbert: "Über das Unendliche" (1925)]

The well-armed state power? After three years already on its last legs?

For original German texts and links to sources see "Das Kalenderblatt 090813". http://www.hs-augsburg.de/~mueckenh/KB/KB%20001-200.pdf

§ 037 Brouwer made it clear, as I think beyond any doubt, that there is no evidence supporting the belief in the existential character of the totality of all natural numbers, and hence the principle of excluded middle in the form "Either there is a number of the given property g, or all numbers have the property Ÿg" is without foundation. [...] The sequence of numbers which grows beyond any stage already reached by passing to the next number, is a manifold of possibilities open towards infinity; it remains forever in the status of creation, but is not a closed realm of things existing in themselves. That we blindly converted one into the other is the true source of our difficulties, including the - a source of more fundamental nature than Russell's vicious circle principle indicated ["No totality can contain members defined in terms of itself"]. Brouwer opened our eyes and made us see how far classical mathematics, nourished by a belief in the "absolute" that transcends all human possibilities of realization, goes beyond such statements as can claim real meaning and truth founded on evidence. According to this view and reading of history, was abstracted from the mathematics of finite sets and their subsets. (The word finite is here to be taken in the precise sense that the members of such set are explicitly exhibited one by one.) Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and Original sin of set theory even if no paradoxes result from it. Not that contradictions showed up is surprising, but that they showed up at such a late stage of the game! [Hermann Weyl: "Mathematics and logic: A brief survey serving as a preface to a review of The Philosophy of ", American Mathematical Monthly 53 (1946) 2-13] [Komaravolu Chandrasekharan: "Hermann Weyl, Gesammelte Abhandlungen, Vol. IV", Springer (1968) p. 275f] http://books.google.de/books?id=lNPriL6kG7AC&printsec=frontcover#v=onepage&q&f=false

§ 038 One of the most often heard arguments in favour of transfinite set theory is the completeness requirement of — and real functions. It is not true.

Well, then tell me, Herr Professor Doktor Mueckenheim, how do you solve the equation ih(∑u/∑t) = H(u) [JR, Matheology § 022, sci.logic, June 13, 2012]

In general most real numbers lack names, and we cannot effectively distinguish them. [AS, Matheology § 022, sci.logic, June 13, 2012]

Numbers are free creations of human mind. They serve as a means to easen and to sharpen the perception of the differences of things. Zahlen sind freie Schöpfungen des menschlichen Geistes, sie dienen als ein Mittel, um die Verschiedenheit der Dinge leichter und schärfer aufzufassen." [: "Was sind und was sollen die Zahlen?" 1887, 8. Aufl. Vieweg, Braunschweig 1960, p. III]

But if not even the numbers can be distinguished, what are they good for? Real numbers that cannot be distinguished will not complete mathematics, they will not make the real axis continuous, they cannot guarantee that every polynomial has its zeros. All real numbers that ever can appear in mathematical calculations have finite names (at least the definition of the problem, like: "find the fourth root of 16") and belong to a countable set. Therefore uncountably many unreal real numbers are good for nothing.

I am convinced that the platonism which underlies Cantorian set theory is utterly unsatisfactory as a philosophy of our subject [...] platonism is the medieval metaphysics of mathematics; surely we can do better. [S. Feferman: "Infinity in Mathematics: Is Cantor Necessary?"]

Feferman shows in his article, "Why a little bit goes a long way. Logical foundations of scientifically applicable mathematics" on the basis of a number of case studies that the mathematics currently required for scientific applications can all be carried out in an axiomatic system whose basic justification does not require the actual infinite. http://www.hs-augsburg.de/~mueckenh/GU/GU11.PPT#416,62,Folie 62

"The actual infinite is not required for the mathematics of the physical world." [S. Feferman: "In the light of logic", Oxford Univ. Press (1998) p. 30] http://books.google.de/books?id=AadVrcnschMC&pg

Though Gödel has been identified as the leading defender of set-theoretical platonism, surprisingly even he at one point regarded it as unacceptable. In his concluding chapters, Feferman uses tools from the special part of logic called proof theory to explain how the vast part if not all of scientifically applicable mathematics can be justified on the basis of purely arithmetical principles. At least to that extent, the question raised in two of the essays of the volume, "Is Cantor Necessary?", is answered with a resounding "no". [S. Feferman, loc. cit, description from the jacket flap] http://math.stanford.edu/~feferman/book98.html

§ 039 Cantor devised set theory for application to reality. In a letter to Hilbert he wrote about his plan of a paper on set theory and its applications:

The third part contains the applications of set theory to the natural sciences: physics, chemistry, mineralogy, botany, zoology, anthropology, biology, physiology, medicine etc. It is what Englishmen call "natural philosophy". In addition we have the so called "humanities", which, in my opinion, have to be called natural sciences too, because also the "mind" belongs to nature. Der dritte Theil bringt die Anwendungen der Mengenlehre auf die Naturwissenschaften: Physik, Chemie, Mineralogie, Botanik, Zoologie, Anthropologie, Biologie, Physiologie, Medizin etc. Ist also das, was die Engländer "Natural philosophy" nennen. Dazu kommen aber auch Anwendungen auf die sogenannten "Geisteswissenschaften", die meines Erachtens als Naturwissenschaften aufzufassen sind; denn auch der "Geist" gehört mit zur Natur. [G. Cantor to D. Hilbert, letter of September 20, 1912]

In a letter to Mittag-Leffler Cantor explained his impetus for devising set theory:

Further I am busy with scrutinizing the applications of set theory to the physiology of organisms. [...] I have been occupied for 14 years with these ideas of a closer exploration of the basic nature of all organic; they are the true reason why I have undertaken the painstaking and hardly rewarding business of investigating point sets, and all the time never lost sight of it, not for a moment. Further I am interested, purely theoretically, in the nature of the states and what belongs to them, because I have my opinions on that topic which later may become formulated mathematically; perhaps you will be taken aback, but that striking impression will disappear, when you consider that also the state in some sense represents an organic being. Ausserdem bin ich mit Untersuchungen über Anwendungen der Mengenlehre auf die Naturlehre der Organismen beschäftiget [...] Mit diesen Ideen einer genaueren Ergründung des Wesens alles Organischen beschäftige ich mich schon seit 14 Jahren, sie bilden die eigentliche Veranlassung, weshalb ich das mühsame und wenig Dank verheissende Geschäft der Untersuchung von Punctmengen unternommen und in diesem Zeitraum keinen Augenblick aus den Augen verloren habe. Ausserdem interessirt mich rein theoretisch das Wesen des Staates und was dazu gehört, weil ich auch darüber meine Gesichtspuncte habe, die zu mathematischer Formulierung später führen dürften; das Auffallende, was Sie darin vielleicht finden, verschwindet, wenn sie erwägen, dass auch der Staat ein organisches Wesen gewissermaassen repräsentirt. [G. Cantor to G. Mittag-Leffler, Sept. 22, 1884]

By applied set theory I understand what usually is called natural science or . To this realm belong completely all natural sciences, those concerning the anorganic as well as the organic world. Unter angewandter Mengenlehre verstehe ich Dasjenige, was man Naturlehre oder Kosmologie zu nennen pflegt, wozu also die sämmtlichen sogenannten Naturwissenschaften gehören, sowohl die auf die anorganische, wie auch auf die organische Welt sich beziehenden. [Ivor Grattan-Guinness: "An unpublished paper by Georg Cantor: Principien einer Theorie der Ordnungstypen. Erste Mittheilung.", Acta Mathematica 124 (1970) 65-107]

I have held the following hypothesis for years: The cardinality of the body-matter is what I call, in my investigations, the first cardinality, the cardinality of the ether-matter, on the other hand, is the second. ... habe ich mir schon vor Jahren die Hypothese gebildet, daß die Mächtigkeit der Körpermaterie diejenige ist, welche ich in meinen Untersuchungen die erste Mächtigkeit nenne, daß dagegen die Mächtigkeit der Äthermaterie die zweite ist. [G. Cantor: "Ueber verschiedene Theoreme aus der Theorie der Punktmengen in einem n-fach ausgedehnten stetigen Raume Gn. Zweite Mitteilung.", Acta Mathematica Vol. 7, p. 105-124 (1885)]

Today we know that there is absolutely no application of infinite set theory to reality, simply because there is no actual infinity in reality.

§ 040

Does the infinitely small exist in reality? Quarks are the smallest elementary particles presently known. Down to 10-19 m there is no structure detectable. Many physicists including the late W. Heisenberg are convinced that there is no deeper structure of matter. On the other hand, the with molecules, atoms, and elementary particles suggests that these physicists may be in error and that matter may be further divisible. However, it is not divisible in infinity. There is a clear-cut limit. Lengths which are too small to be handled by material meter sticks can be measured in terms of wavelengths λ of electromagnetic waves, for instance. λ = c/ν (c = 3ÿ108 m/s) The frequency ν is given by the E of the photon ν = E/h (h = 6,6ÿ10-34 Js) and a photon cannot contain more than all the energy of the universe E = mÿc2 which has a mass of about m = 5ÿ1055 g. This yields the complete energy E = 5ÿ1069 J. So the unsurpassable minimal length is 4ÿ10-95 m.

Does the infinitely large exist in reality? Modern cosmology teaches us that the universe has a beginning and is finite. But even if we do not trust in this wisdom, we know that theory of relativity is as correct as human knowledge can be. According to relativity theory, the accessible part of the universe is a sphere of 50ÿ109 LY radius containing a volume of 1080 m3. (This sphere is growing with time but will remain finite forever.) "Warp" propulsion, "worm hole" traffic, and other science fiction (and scientific fiction) does not work without time reversal. Therefore it will remain impossible to leave (and to know more than) this finite sphere. Modern quantum mechanics has taught us that entities which are non-measurable in principle, do not exist. Therefore, also an upper bound (which is certainly not the supremum) of 10365 for the number of elementary spatial cells in the universe can be calculated from the minimal length estimated above.

[W. Mückenheim: "The infinite in sciences and arts", Proc. 2nd Intern. Symp. of Mathematics and its Connections to the Arts and Sciences (MACAS 2), B. Sriraman, C. Michelsen, A. Beckmann, V. Freiman (eds.), Centre for Science and Mathematics Education, University of Southern Denmark, Odense 2008, p. 265 - 272] http://arxiv.org/abs/0709.4102

§ 041 Aristotle is the first to distinguish potential infinity and actual infinity. He bans actual infinity from philosophy and mathematics. The idea of the infinity of God, created in Hellenism, amalgamates - not later than in the works of Thomas Aquinatus - with the Aristotelian postulate of the pure actuality of God. This yields the Christian perception of God's pure actuality. During the renaissance, in particular with Bruno, the actual infinity is carried over from God to the world. The finite world models of present science show clearly, how the superiority of the idea of actual infinity has ceased, caused by the classical (modern) physics. In contrast it appears disconcerting that G. Cantor explicitly established the actual infinity in mathematics during the end of the last century. In the intellectual framework of our century - in particular when considering existential philosophy - the actual infinity appears just like an anachronism. Aristoteles unterscheidet als erster das Potentiell-Unendliche vom Aktual-Unendlichen - und verbannt das Aktual-Unendliche aus der Philosophie und Mathematik. Der Gedanke der Unendlichkeit Gottes, der aus dem Hellenismus stammt, verbindet sich - spätestens bei Thomas - mit der von Aristoteles postulierten reinen Aktualität Gottes. So entsteht die christliche Auffassung Gottes als aktualer Unendlichkeit. In der Renaissance, besonders bei Bruno, überträgt sich die aktuale Unendlichkeit von Gott auf die Welt. Die endlichen Weltmodelle der gegenwärtigen Naturwissenschaft zeigen deutlich, wie diese Herrschaft des Gedankens einer aktualen Unendlichkeit mit der klassischen (neuzeitlichen) Physik zu Ende gegangen ist. Befremdlich wirkt dem gegenüber die Einbeziehung des Aktual-Unendlichen in die Mathematik, die explizit erst gegen Ende des vorigen Jahrhunderts mit G. Cantor begann. Im geistigen Gesamtbilde unseres Jahrhunderts - insbesondere bei Berücksichtigung des existenzialistischen Philosophierens - wirkt das Aktual-Unendliche geradezu anachronistisch. [Paul Lorenzen: "Das Aktual-Unendliche in der Mathematik", Philosophia naturalis 4 (1957) 3-11] http://www.sgipt.org/wisms/geswis/mathe/ulorenze.htm#Das Aktual-Unendliche in der Mathematik

§ 042 Recent history demonstrates that anyone foolhardy enough to describe his own work as "rigorous" is headed for a fall. Therefore, we shall claim only that we do not knowingly give erroneous arguments. We are conscious also of writing for a large and varied audience, for most of whom clarity of meaning is more important than "rigor" in the narrow mathematical sense. There are two more, even stronger reasons for placing our primary emphasis on logic and clarity. Firstly, no argument is stronger than the that go into it, and as Harold Jeffreys noted, those who lay the greatest stress on mathematical rigor are just the ones who, lacking a sure sense of the real world, tie their arguments to unrealistic premises and thus destroy their relevance. Jeffreys likened this to trying to strengthen a building by anchoring steel beams into plaster. An argument which makes it clear intuitively why a result is correct, is actually more trustworthy and more likely of a permanent place in science, than is one that makes a great overt show of mathematical rigor unaccompanied by understanding. Secondly, we have to recognize that there are no really trustworthy standards of rigor in a mathematics that has embraced the theory of infinite sets. Morris Kline (1980, p. 351) came close to the Jeffreys simile: "Should one design a bridge using theory involving infinite sets or the axiom of choice? Might not the bridge collapse?" The only real rigor we have today is in the operations of elementary arithmetic on finite sets of finite integers, and our own bridge will be safest from collapse if we keep this in mind. [...] Finally, some readers should be warned not to look for hidden subtleties of meaning which are not present. [...] There are no linguistic tricks and there is no "meta-language" gobbledygook; only plain English. We think that this will convey our message clearly enough to anyone who seriously wants to understand it. In any , we feel sure that no further clarity would be achieved by taking the first few steps down that infinite regress that starts with: "What do you mean by 'exists'?" [E. T. Jaynes: "Probability Theory: The Logic of Science", (Fragmentary Edition of March 1996)] http://www-biba.inrialpes.fr/Jaynes/cpreambl.pdf http://de.wikipedia.org/wiki/Edwin_Thompson_Jaynes

§ 043 Pure mathematics and science are finally being reunited and, mercifully, the Bourbaki plague is dying out. [Murray Gell-Mann: "Nature Conformable to Herself", Bulletin of the Santa Fe Institute, 7 (1992) 7-10]

§ 044 Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap.

The Jacobi (which forces the heights of a triangle to cross at one point) is an experimental fact in the same way as that the Earth is round (that is, homeomorphic to a ball). But it can be discovered with less expense.

In the middle of the twentieth century it was attempted to divide physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences. They first began teaching their ugly scholastic pseudo-mathematics to their students, then to schoolchildren (forgetting Hardy's warning that ugly mathematics has no permanent place under the Sun).

Since scholastic mathematics that is cut off from physics is fit neither for teaching nor for application in any other science, the result was the hate towards mathematicians - both on the part of the poor schoolchildren (some of whom in the meantime became ministers) and of the users.

The ugly building, built by undereducated mathematicians who were exhausted by their inferiority complex and who were unable to make themselves familiar with physics [...] predominated in the teaching of mathematics for decades. Having originated in France, this pervertedness quickly spread to teaching of foundations of mathematics, first to university students, then to school pupils of all lines (first in France, then in other countries, including Russia).

To the question "what is 2 + 3" a French primary school pupil replied: "3 + 2, since addition is commutative". He did not know what the sum was equal to and could not even understand what he was asked about!

Another French pupil (quite rational, in my opinion) defined mathematics as follows: "there is a square, but that still has to be proved".

Judging by my teaching experience in France, the university students' idea of mathematics (even of those taught mathematics at the École Normale Supérieure - I feel sorry most of all for these obviously intelligent but deformed kids) is as poor as that of this pupil.

For example, these students have never seen a paraboloid and a question on the form of the surface given by the equation xy = z2 puts the mathematicians studying at ENS into a stupor. Drawing a curve given by parametric equations (like x = t3 - 3t, y = t4 - 2t2) on a plane is a totally impossible problem for students (and, probably, even for most French professors of mathematics).

[V.I. Arnold: "On teaching mathematics" (1997), Mathematics in Palais de Découverte in Paris on 7 March 1997, Translated by A.V. Goryunov] http://pauli.uni-muenster.de/~munsteg/arnold.html

§ 045 This question brings to the fore something that is fundamental and pervasive: that what we are doing is finding ways for people to understand and think about mathematics.

The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics.

We have a facility for thinking about processes or sequences of actions that can often be used to good effect in mathematical reasoning. One way to think of a function is as an action, a process, that takes the domain to the range. This is particularly valuable when composing functions. Another use of this facility is in remembering proofs: people often remember a proof as a process consisting of several steps. In topology, the notion of a homotopy is most often thought of as a process taking time. Mathematically, time is no different from one more spatial dimension, but since humans interact with it in a quite different way, it is psychologically very different.

On the most fundamental level, the foundations of mathematics are much shakier than the mathematics that we do. Most mathematicians adhere to foundational principles that are known to be polite fictions. For example, it is a theorem that there does not exist any way to ever actually construct or even define a well-ordering of the real numbers. There is considerable evidence (but no proof) that we can get away with these polite fictions without being caught out, but that doesn’t make them right. Set theorists construct many alternate and mutually contradictory “mathematical universes” such that if one is consistent, the others are too. This very little confidence that one or the other is the right choice or the natural choice.

[Bill Thurston: "On proof and progress in mathematics", Bull. of the American Math. Soc. 30, 2, (1994) 161-177] http://arxiv.org/PS_cache/math/pdf/9404/9404236v1.pdf http://www.hs-augsburg.de/~mueckenh/GU/GU11c

§ 046 If a non-terminating decimal is to be handled or arranged in sequence like a thing it is sufficient to know how to handle and arrange a finite decimal of n digits, the number n being subject to no restriction as to magnitude. The theorem would now demand that it is impossible to set up any scheme for arranging all possible decimal fractions of n digits in a definite order, n being subject to no restriction as to magnitude. But such a theorem is obviously false, for there are 10n possible decimals of n digits [...] What is done in the actual diagonal Verfahren when translated into this technique is this: it is shown that given a proposed array and any number n, no matter how large, it is then possible to set up a decimal the first n digits of which are different from the first n digits of any decimal to be found in the first n places of the proposed array. But this is clearly not what is required.

The ordinary diagonal Verfahren I believe to involve a patent confusion of the program and object aspects of the decimal fraction, which must be apparent to any who imagines himself actually carrying out the operations demanded in the proof. In fact, I find it difficult to understand how such a situation should have been capable of persisting in mathematics. Doubtless the confusion is bound up with the notion of existence; the decimal fractions are supposed to "exist" whether they can be actually produced and exhibited or not. But from the operational point of view all such notions of "existence" must be judged to be obscured with a thick metaphysical haze, and to be absolutely meaningless from the point of view of those restricted operations which can be allowed in mathematical inquiry. This repudiation of the conventional proof by the diagonal Verfahren of the non-denumerability of the non-terminating decimals will be found to be very similar in spirit, although not in detail, to the argument in Bentley's book [Linguistic Analysis of Mathematics]. It may be worth while to record that the argument above was reached by me independently of Bentley [...] One can obviously say that all the rules for writing down nonterminating decimals formulatable by the entire human race up to any epoch in the future must be denumerable [...] I do not know what it means to talk of numbers existing independent of the rules by what they are determined; operationally there is nothing corresponding to the concept. If it means anything to talk about the existence of numbers, then there must be operations for determining whether alleged numbers exist or not, and in testing the existence of a number how shall it be identified except by means of the rules? [P.W. Bridgman: "A physicist's second reaction to Mengenlehre", Scripta Mathematica, Vol. II, 1934]

§ 047 Occasionally logicians inquire as to whether the current "Axioms" need to be changed further, or augmented. The more fundamental question - whether mathematics requires any Axioms - is not up for discussion. That would be like trying to get the high on the island of Okineyab to consider not whether the Divine Ompah's Holy Phoenix has twelve or thirteen colours in her tail (a fascinating question on which entire tomes have been written), but rather whether the Divine Ompah exists at all. Ask that question, and icy stares are what you have to expect, then it's off to the dungeons, mate, for a bit of retraining. Mathematics does not require "Axioms". The job of a pure mathematician is not to build some elaborate castle in the sky, and to proclaim that it stands up on the strength of some arbitrarily chosen assumptions. The job is to investigate the mathematical reality of the world in which we live. For this, no assumptions are necessary. Careful observation is necessary, clear definitions are necessary, and correct use of language and logic are necessary. But at no point does one need to start invoking the existence of objects or procedures that we cannot see, specify, or implement. The difficulty with the current reliance on "Axioms" arises from a grammatical confusion [...] People use the term "Axiom" when often they really mean definition. Thus the "axioms" of group theory are in fact just definitions. We say exactly what we mean by a group, that's all. [..] Euclid may have called certain of his initial statements Axioms, but he had something else in mind. Euclid had a lot of geometrical facts which he wanted to organize as best as he could into a logical framework. Many decisions had to be made as to a convenient order of presentation. He rightfully decided that simpler and more basic facts should appear before complicated and difficult ones. So he contrived to organize things in a linear way, with most Propositions following from previous ones by logical reasoning alone, with the exception of certain initial statements that were taken to be self-evident. To Euclid, an Axiom was a fact that was sufficiently obvious to not require a proof. This is a quite different meaning to the use of the term today. Those formalists who claim that they are following in Euclid's illustrious footsteps by casting mathematics as a game played with symbols which are not given meaning are misrepresenting the situation. [...] And yes, all right, the Continuum hypothesis doesn't really need to be true or false, but is allowed to hover in some no-man's land, falling one way or the other depending on what you believe. Cohen's proof of the independence of the Continuum hypothesis from the "Axioms" should have been the long overdue wake-up call. In ordinary mathematics, statements are either true, false, or they don't make sense. If you have an elaborate theory of "hierarchies upon hierarchies of infinite sets", in which you cannot even in principle decide whether there is anything between the first and second "infinity" on your list, then it's time to admit that you are no longer doing mathematics. Whenever discussions about the foundations of mathematics arise, we pay lip service to the "Axioms" of Zermelo-Fraenkel, but do we ever use them? Hardly ever. With the notable exception of the "Axiom of Choice", I bet that fewer than 5% of mathematicians have ever employed even one of these "Axioms" explicitly in their published work. The average mathematician probably can't even remember the "Axioms". I think I am typical - in two weeks time I'll have retired them to their usual spot in some distant ballpark of my memory, mostly beyond recall. [...] Do you really think you need to have all the natural numbers together in a set to define the function on natural numbers? Of course not - the rule itself, together with the specification of the kinds of objects it inputs and outputs is enough. As computer scientists already know. [N J Wildberger: "Set Theory: Should You Believe?"] http://web.maths.unsw.edu.au/~norman/views2.htm

§ 048 The author [A. A. Fraenkel] is well known for his research in set theory as well as his published textbooks in this subject. He has previously written the book "Einleitung in die Mengenlehre" which appeared in three editions. The last edition which was published in 1928 was reprinted in New York in 1946. Whereas "Einleitung in die Mengenlehre" contained an exposition of classical set theory as well as a survey of modern theoretical research in the foundations of mathematics, Fraenkel has now decided to write the present book as an account of the classical theory only. The modern aspects of foundation theory will be discussed in another book under the title "Foundations of Set Theory" which is due to appear about 1955. Presumably, the reason for this division of the contents of "Einleitung in die Mengenlehre" into two different books is that the subject matter has grown too large. The reviewer, however, is not enthusiastic about this division since such a textbook as the present one will be read primarily by students and they might form the impression that classical set theory is securely founded just as other parts of mathematics, e.g. arithmetic. Such an impression would, however, be misleading. If it were not so, we could omit the entire modern foundational research without real loss to mathematics. To the reviewer it seems unfortunate that classical set theory is developed in a separate book so that all scruples - or almost all of them - are reserved for the second volume. This might have the effect that most readers of this present volume will probably not become acquainted with the criticisms at all. It is true that some hints to such scruples are given, but most students might not think that they are important. On the other hand, it must be conceded that the lack of knowledge of the results of foundational research will not mean much to mathematicians who are not especially interested in the logical development of mathematics. [Th. Skolem: "Review of: A. A. Fraenkel : Abstract Set Theory. Amsterdam & Groningen, North- Holland Company, 1953. XII + 479 pp." Mathematica Skandinavica 1 (1953) 313.] http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=179577

§ 049 How can a matheologian be distinguished from a mathematician? A witches' ordeal.

Take a circle (it need not be glowing) and ask the examinee to mark infinitely many intervals by infinitely many endpoints. Shuffle the endpoints such that they slide along the circle in a completely arbitrary way. Ask again how many endpoints are there and how many intervals these endpoints define. The answer of a mathematician will be "infinitely many" in both cases. The answer of a matheologian will be split. Yes, there are infinitely many endpoints, but between them there are uncountably many (that is more than infinitely many) intervals, mainly degenerate intervals, so called singletons: Between two endpoints there is only one point. However, the endpoints limiting the singleton, cannot be identified and therefore cannot increase the number of endpoints such that it is the same as the number of singletons. If you ask what makes the singletons singletons, you will be told that there are infinitely many end- and other points, far remote from the singleton, that stabilize the situation like the signs of the zodiac stabilize the human fate and let all of us act as we do. Under this aspect even we could boast to be uncountably many.

Well that's the matheologians' ordeal. If you, a reader of sci.math, meet a matheologian, try to escape before he has caught you and turned your brain upside down.

§ 050 Already during Cantor’s life time, the reception of his ideas was more like that of new trends in the art, such as impressionism or atonality, than that of new scientific theories. It was highly emotionally charged and ranged from total dismissal (Kronecker’s “corrupter of youth”) to highest praise (Hilbert’s defense of “Cantor’s Paradise”). (Notice however the commonly overlooked nuances of both statements which subtly undermine their ardor: Kronecker implicitly likens Cantor to Socrates, whereas Hilbert with faint mockery hints at Cantor’s conviction that Set Theory is inspired by God.) [Yuri I. Manin: Georg Cantor and his heritage (2002)] http://aps.arxiv.org/PS_cache/math/pdf/0209/0209244v1.pdf

§ 051 Herren Geheimrat Hilbert und Prof. Dr. Cantor, I'd like to be Excused from your "Paradise": It is a Paradise of Fools, and besides feels more like Hell famously said: "No one shall expel us from the paradise that Cantor has created for us." Don't worry, dear David and dear Georg, I am not trying to kick you out. But, it won't be quite as much fun, since you won't have the pleasure of my company. I am leaving on my own volition. For many years I was sitting on the fence. I knew that it was a paradise of fools, but so what? We humans are silly creatures, and it does not harm anyone if we make believe that ¡0, ¡1, etc. have independent existence. Granted, some of the greatest minds, like Gödel, were fanatical platonists and believed that infinite sets existed independently of us. But if one uses the name- dropping rhetorics, then one would have to accept the veracity of Astrology and , on the grounds that Newton and Kepler endorsed them. An equally great set theorist, Paul Cohen, knew that it was only a game with axioms. In other words, Cohen is a sincere formalist, while Hilbert was just using formalism as a rhetoric sword against intuitionism, and deep in his heart he genuinely believed that Paradise was real. My mind was made up about a month ago, during a wonderful talk (in the Integers 2005 conference in honor of Ron Graham's 70th birthday) by MIT (undergrad!) Jacob Fox (whom I am sure you would have a chance to hear about in years to come), that meta-proved that the answer to an extremely concrete question about coloring the points in the plane, has two completely different answers (I think it was 3 and 4) depending on the axiom system for Set Theory one uses. What is the right answer?, 3 or 4? Neither, of course! The question was meaningless to begin with, since it talked about the infinite plane, and infinite is just as fictional (in fact, much more so) than white unicorns. Many , it works out, and one gets seemingly reasonable answers, but Jacob Fox's example shows that these are flukes. It is true that the Hilbert-Cantor Paradise was a practical necessity for many years, since humans did not have computers to help them, hence lots of combinatorics was out of reach, and so they had to cheat and use abstract nonsense, that Paul Gordan rightly criticized as theology. But, hooray!, now we have computers and combinatorics has advanced so much. There are lots of challenging finitary problems that are just as much fun (and to my eyes, much more fun!) to keep us busy. Now, don't worry all you infinitarians out there! You are welcome to stay in your Paradise of fools. Also, lots of what you do is interesting, because if you cut-the-semantics-nonsense, then you have beautiful combinatorial structures, like John Conway's surreal numbers that can "handle" "infinite" ordinals (and much more beyond). But as Conway showed so well (literally!) it is "only" a (finite!) game. While you are welcome to stay in your Cantorian Paradise, you may want to consider switching to my kind of Paradise, that of finite combinatorics. No offense, but most of the infinitarian lore is sooo boring and the Bourbakian abstract nonsense leaves you with such a bitter taste that it feels more like Hell. But, if you decide to stick with Cantor and Hilbert, I will still talk to you. After all, eating meat is even more ridiculous than believing in the (actual) infinity, yet I still talk to carnivores, (and even am married to one). [Doron Zeilberger: "Opinion 68" (2005)] http://www.math.rutgers.edu/~zeilberg/Opinion68.html

§ 052 Most of the debate on the internet about Cantor's Theory is junk. The topic is a crank magnet. Most of the people who participate in the debate, have no deep understanding of the issues. However, hidden within all the noise, there does seem to be some signal. While the pure mathematicians almost unanimously accept Cantor's Theory (with the exception of a small group of constructivists), there are lots of intelligent people who believe it to be an . Typically, these people are non-experts in pure mathematics, but they are people who have found mathematics to be of great practical value in science and technology, and who like to view mathematics itself as a science. These "anti-Cantorians" see an underlying reality to mathematics, namely, computation. They tend to accept the idea that the computer can be thought of as a microscope into the world of computation, and mathematics is the science which studies the phenomena observed through that microscope. They claim that that paradigm encompasses all of the mathematics which has the potential to be applied to the task of understanding phenomena in the real world (e.g. in science and engineering). Cantor's Theory, if taken seriously, would lead us to believe that while the collection of all objects in the world of computation is a countable set, and while the collection of all identifiable abstractions derived from the world of computation is a countable set, there nevertheless "exist" uncountable sets, implying (again, according to Cantor's logic) the "existence" of a super-infinite fantasy world having no connection to the underlying reality of mathematics. The anti-Cantorians see such a belief as an absurdity (in the sense of being disconnected from reality, rather than merely counter-intuitive). The mathematicians claim that they can "prove" the existence of uncountable sets, and hence there's nothing to be debated. But that merely calls into question the nature of "proof". Certainly infinite sets and power sets exist as abstractions. But, abstractions don't necessarily obey exactly that same laws of logic as directly observable objects. Assuming otherwise can turn abstractions into fantasies, and proofs into , and that's the crux of the anti-Cantorian's argument. The pure mathematicians tend to view mathematics as an art form. They seek to create beautiful theories, which may happen to be connected to reality, but only by accident. Those who apply mathematics, tend to view mathematics as a science which explores an objective reality (the world of computation). In science, truth must have observable implications, and such a "reality check" would reveal Cantor's Theory to be a ; many of the formal theorems in Cantor's Theory have no observable implications. The artists see the requirement that mathematical statements must have observable implications as a restriction on their intellectual freedom. The "anti-Cantorian" view has been around ever since Cantor introduced his ideas. [...] In the contemporary mainstream mathematical literature, there is almost no debate over the validity of Cantor's Theory. [...] It was the advent of the internet which revealed just how prevalent the anti- Cantorian view still is; there seems to be a never-ending heated debate in the Usenet newsgroups sci.math and sci.logic over the validity of Cantor's Theory. Typically, the anti- Cantorians accuse the pure mathematicians of living in a dream world, and the mathematicians respond by accusing the anti-Cantorians of being imbeciles, idiots and crackpots. It is plausible that in the future, mathematics will be split into two disciplines - scientific mathematics (i.e. the science of phenomena observable in the world of computation), and philosophical mathematics, wherein Cantor's Theory is merely one of many possible formal "theories" of the infinite. [David Petry, sci.math, sci.logic, 20 Juli 2005] http://groups.google.com/group/sci.logic/msg/02ee220b035488f9?dmode=source

§ 053 It seems reasonable that were platonic in Plato's times, but is certainly surprising the persistence of that primitive way of thinking in the community of contemporary mathematicians [...] But for those of us who believe in the organic nature of our brains and in its abilities of perceiving and knowing modelled through more than 3600 millions years of organic evolution, platonism has no longer sense. And neither self- nor the actual infinity may survive away from the platonic scenario. On the other hand, it seems convenient to recall the long and conflictive history of both notions (would them have been so conflictive if they were consistent?); and above all their absolute uselessness in order to know the natural world. Physics and even mathematics could go without both notions. Experimental sciences as chemistry, biology and geology have never been related to them. The potential infinity probably suffices. Even the number of distinguishable sites in the universe is finite. Finite and discrete: not only matter and energy are discrete entities, space and time could also be of a discrete - quantum - nature as is being suggested from some areas of contemporary physics as superstring theory, loop quantum gravity, euclidean quantum gravity, quantum computation, or black holes thermodynamics. [Antonio Leon Sanchez: "Extending Cantor's " (2012)] http://arxiv.org/abs/0809.2135

§ 054 They squabbled over lots of wine 'bout Darwin's ideas that might humilitate mankind and take the shine of human honour and pride!

They drank some mugs and steins so vast, they stumbled and swayed through the doors. They grunted audibly and crawled home at last through dirt and mud on .

(After Wilhelm Busch, Kritik des Herzens, 1874 (the year when Cantor started set theory)) http://de.wikisource.org/wiki/Sie_stritten_sich_beim_Wein_herum

Well, Husserl is, as I know with absolute certainty and as is clear from his local lectures about proofs of God and against Darwinism, a theist and therefore is better suited as a teacher of catholic students than the favourite candidates of Prof. Riehl. [Cantor to canon Woker, Nov. 30, 1895]

Indeed, I think that the faculties, connect by an untearable relationship, cannot be indifferent towards the question, whether the professor of the philosophical faculty is theist or atheist, whether he acts pro Darwinism or contra Darwinism. I did not want to initiate a direct action of the theological faculty in favour of Husserl, but I meant, supported by Father Schwermer's information about the influence of the theological faculty on the government, that a private, inofficial connivance of the candidate, so warmly recommended by myself, would not be impossible. [Cantor to canon Woker, Dec. 15, 1895]

As you requested I send some information concerning the candidates that presently are under consideration by your philosophical faculty: 1. Siebeck in Gießen (1842) 2. Avenarius in Zürich (1843) 3. Eucken in Jena (1846) 4. Natorp in Marburg (1854) 5. Spitzer in Graz (1854) 6. Gross in Gießen (1861) 7. Busse in Marburg (1862) As much as I would like to recommend the standpoint of Tolerari posse with respect to my young friend Husserl, I have to express my greatest reservations towards those seven names. [...] Ad. 5. Has initially written a book about Darwinism, and nothing after that. Is probably a Jew and radically liberal in every respect. [Cantor to Heiner, Jan. 11, 1896]]

{{In order to reassure the reader's sense of justice I can say that Cantor's intrigues everytime were condemned to fail. German appointment committees act properly according to laws and conscience, as I can say from my own experience. But Darwinism was not the only spiritual enemy that Cantor felt vocated to fight:}}

Now, the question rises whether it will be possible to reach the final aim, namely the total destruction of the living-principle of freemasonry in all its facets. This is the purpose why I have investigated and studied this dragon right through the centre of its black-blooded heart. I believe that on this way I have been lead and furthered by the mercy of God. [Cantor to Hermite, Febr. 11, 1896]

Original German texts can be found in: Das Kalenderblatt 090622 http://www.hs-augsburg.de/~mueckenh/KB/KB%20001-200.pdf

§ 055 Gödel takes the paradoxes very seriously; they reveal to him "the amazing fact that our logical intuitions are self-contradictory." This attitude toward the paradoxes is of course at complete variance with the view of Brouwer who blames the paradoxes not on some transcendental logical intuition which deceives us, but on a gross error inadvertently committed in the passage from finite to infinite sets. I confess that in this respect I remain steadfastly on the side of Brouwer. [H. Weyl: "Philosophy of Mathematics and Natural Science", Princeton, 1949] [Komaravolu Chandrasekharan: "Hermann Weyl, Gesammelte Abhandlungen, Vol. IV", Springer (1968) p. 602] [Hermann Weyl: "Philosophie der Mathematik und Naturwissenschaft", Oldenbourg, 8. Aufl. (2009)] http://www.oldenbourg- verlag.de/search/apachesolr_search/?_suche%25255Bmode%25255D=einfach&sv%25255Bolb _vt%25255D=Weyl

§ 056 The mathematician Frege demolished the more traditional attempts to explain and establish number and mathematical certainty. (To his satisfaction, at any rate). Frege went on to try to give natural numbers and arithmetic a sound rational basis. To him, this meant in part giving them a basis in a scheme of calculation. Frege faced many obstacles. Brouwer, and Wittgenstein in his later period, proclaimed that mathematics cannot be founded by logic, or by another layer of calculations. Frege's work was ignored in Germany; then, when his system was finally published, it was immediately wrecked by Russell's paradox. In spite of this, Frege's program was ultimately upheld by the professional majority (against Brouwer, for example). The mathematical logic elaborated by Frege, and his cohorts Boole, Cantor, Peano, Skolem, Herbrand, Hilbert, etc., became the prevailing view, or contextualization, of elementary arithmetic. In no way is that a trivial remark or outcome. Calling themselves new hens, the foxes seized control of the henhouse. Arithmetic had been reductionistically re-founded in a way which nullified its traditional consistency and uniqueness. By no means did the discovery that Frege's system was inconsistent kill it. Instead, it was repaired and adopted: even though the problem of repair was obdurate, and vitiated the tenet of uniqueness of the natural numbers which crystallized Frege's original goal. (The introduction of and the Axiom of Reducibility.) Thus, the majority was prepared to resort to "scandalously artificial" devices to prop up a system which in its straightforward formulation was inconsistent. - Because the majority loved the new shell game. It must also be said that a significant but weak minority opposed this course, e.g. Brouwer, Heyting, Weyl, Wittgenstein. The lesson is that when an initially unpopular theory became popular, then prima facie inconsistency did not kill it. It was upheld by casuistry, as it were - even though it continued to be opposed by a minority. [Henry Flynt: Is mathematics a scientific discipline? (1994)] http://www.henryflynt.org/studies_sci/mathsci.html

§ 057 I have seen some ultrafinitists go so far as to challenge the existence of 2100 as a natural number, in the sense of there being a series of "points" of that length. There is the obvious "draw the line" objection, asking where in 21, 22, 23, ... , 2100 do we stop having "Platonistic reality"? Here this ... is totally innocent, in that it can be easily be replaced by 100 items (names) separated by commas. I raised just this objection with the (extreme) ultrafinitist Yessenin-Volpin during a lecture of his. He asked me to be more specific. I then proceeded to start with 21 and asked him whether this is "real" or something to that effect. He virtually immediately said yes. Then I asked about 22, and he again said yes, but with a perceptible delay. Then 23, and yes, but with more delay. This continued for a couple of more times, till it was obvious how he was handling this objection. Sure, he was prepared to always answer yes, but he was going to take 2100 times as long to answer yes to 2100 then he would to answering 21. There is no way that I could get very far with this. [Harvey M. Friedman: "Philosophical Problems in Logic"] http://www.math.ohio-state.edu/~friedman/manuscripts.html http://www.math.ohio-state.edu/~friedman/pdf/Princeton532.pdf

§ 058 The only possible conclusion [given the Löwenheim-Skolem Theorem] seems to be that notions such as countablility and uncountability are inherently relative. [...] Our description of P(ω) as uncountable, even though correct, must be understood relative to our own current point of view. From another point of view this very set may be countable. But I want to argue that such , compelling though it is, is subject to the by now familiar predicament that any statement of it, if it is to be intelligible at all, will have to be understood within a framework that casts it as a straighforward error. It is this which I take to be Skolem's paradox. The crux of the matter is this. If there is an implicit relativization in our claim that P(ω) is uncountable (the claim which is established by Cantor's argument), then it ought to be possible to make it explicit (just as it is possible to make explicit any relativization in the claim that a is moving). But it is possible to do so this only insofar as it is possible to construe our discourse about sets as discourse about a particular collection of objects, the collection to which such claims must be relativized. And this in turn is not possible unless we endorse the fundamental error that there is a set which contains all the sets we intend to talk about. When it is claimed that P(ω) is not unconditionally uncountable, we have no way of understanding this except as the demonstrably false claim that it is not uncountable at all. Admittedly, there are interpretations of the language of set theory under which all the "right" sentences come out true [1]

All that depends on context and interpretation (the correct, of course):

"There's - no - con - tra - dic - tion!"

Sometimes I ask myself: is it only accidentally, or is it due to my personality, that this battle shout again and again reminds me of Orwell's cattle shout:

"Four - legs - good - two - legs - bad!"

[2] (Or was it the other way round? I don't remember exactly. I read the book before 1984 already.) and those who are shouting there?

[1] A. W. Moore: "Set Theory, Skolem's Paradox and the Tractatus", Analysis 45 (1985) 13-20. http://analysis.oxfordjournals.org/cgi/pdf_extract/45/1/13 http://www.logicmuseum.com/cantor/skolem_moore.htm [2] George Orwell: "Animal Farm" http://en.wikipedia.org/wiki/Animal_Farm

§ 059 The physical limits to computation have been under active scrutiny over the past decade or two, as theoretical investigations of the possible impact of quantum mechanical processes on computing have begun to make contact with realizable experimental configurations. We demonstrate here that the observed acceleration of the Universe can produce a universal limit on the total amount of information that can be stored and processed in the future, putting an ultimate limit on future technology for any civilization, including a time-limit on Moore's Law. The limits we derive are stringent, and include the possibilities that the computing performed is either distributed or local. A careful consideration of the effect of horizons on information processing is necessary for this analysis, which suggests that the total amount of information that can be processed by any observer is significantly less than the Hawking-Bekenstein entropy associated with the existence of an event horizon in an accelerating universe. [Lawrence M. Krauss, Glenn D. Starkman: "Universal Limits on Computatio" (2004)] http://aps.arxiv.org/abs/astro-ph/0404510

Mathematicians work with abacuses https://www.google.de/search?q=abacuses&hl=de&prmd=imvns&tbm=isch&tbo=u&source=univ &sa=X&ei=USP1T9O0Ac7otQalkr3UBQ&sqi=2&ved=0CHIQsAQ&biw=855&bih=582 or brains, or computers (ABC). Everything beyond that scope - finished infinities, undefinable definitions, and possibility-proofs of actions that provably are impossible to perform - is matheology. Mathematics is what mathematicians do - not what they cannot do.

Enlightenment was a desire for human affairs to be guided by rationality rather than by faith, superstition, or revelation; a belief in the power of human reason to change society and liberate the individual from the restraints of custom or arbitrary authority; all backed up by a world view increasingly validated by science rather than by religion or tradition. [Dorinda Outram: "The Enlightenment", Cambridge University Press, Cambridge (1995)]

It is deeply deplorable that 200 years after that glorious victory of human spirit, the intellectual counter-revolution is still alive.

§ 060 The cardinal contradiction is simply this: Cantor has a proof that there is no greatest cardinal, and yet there are properties (such as "x = x") which belong to all entities. Hence the cardinal number of entities having a property must be the greatest of cardinal numbers. Hence a contradiction [1, p. 31]

An existent class is a class having at least one member. [1, p. 47] {{Surely you are joking Mr. Russell? The class without any member is not among the existent classes?}}

Whether it is possible to rescue more of Cantor's work must probably remain doubtful until the fundamental logical notions employed are more thoroughly understood. And whether, in particular, Zermelo's axiom {{of choice}} is true or false {{I am shocked! An axiom could be true or false in your age, Mr. Russell? Mathematicians in fact tried to find truth and meaning in mathematics?}} is a question which, while more fundamental matters are in doubt, is very likely to remain unanswered. The complete solution of our difficulties, we may surmise, is more likely to come from clearer notions in logic than from the technical advance of mathematics; but until the solution is found we cannot be sure how much of mathematics it will leave intact. [1, p 53] {{O had Mr. Cantor never decided to become a mathematician!}}

Note added February 5th, 1906. - From further investigation I now feel hardly any doubt that the no-classes theory affords the complete solution of all the difficulties stated in the first section of this paper. [1, p 53] {{The classless society has been favoured all over the world at those times. In fact, not much has remained: Cuba and North-Korea. But the no-classes theory is attractive until this very day, because:}} The objections to the theory are [...] that a great part of Cantor's theory of the transfinite, including much that it is hard to doubt, is, so far as can be seen, invalid if there are no classes or . [1, p. 45] {{Is there anything that it is hard to doubt in Cantor's theory?}}

The solution of the difficulties which formerly surrounded the mathematical infinite is probably the greatest achievement of which our age has to boast. [2]

This is an instance of the amazing power of desire in blinding even very able men to fallacies which would otherwise be obvious at once. [3]

[1] Bertrand Russell: "On some difficulties in the theory of transfinite numbers and order types", Proc. London Math. Soc. (2) 4 (1906) 29-53, Received November 24th, 1905. - Read December 14, 1905 http://books.google.de/books/about/On_Some_Difficulties_in_the_Theory_of_Tr.html?id=vczfGw AACAAJ&redir_esc=y [2] Bertrand Russell: “The Study of Mathematics”. New Quarterly, Nov. 1907, 29-44. Reprinted in B. Russell: "Philosophical Essays", Longmans, Green, London" (1910) also B. Russell: "Mysticism and Logic and Other Essays" Longmans, Green & Co, London (1918). Reprinted by Unwin, Paperbacks (1986) [3] Bertrand Russell: "What I believe" from "Why I am not a Christian" (1957) p. 42. Bertrand Russell: "Why I Am Not A Christian and Other Essays on Religion and Related Subjects", (Paul Edwards, ed.), London: George Allen & Unwin (1957) http://www.torrentz.com/4cfdc196f49eb34ea283163acfd1f79dffd667d1 http://en.wikipedia.org/wiki/Bertrand_Russell

061 Hilbert's Hotel is not a paradox, it is a very bad logical mistake, from the first paragraph. It is based on the same terrible mistake that underlies all transfinite math. The mistake is believing that the word "transfinite" can mean something. What it means in practice is really "transinfinite". Mathematicians believe that something can exist beyond infinity. If you accept the addition of 1 to infinity, then it means that you don't understand infinity to begin with. All the math that takes place in the transinfinite is quite simply false. Notice that I do not say it is physically baseless, or mystical, or avant garde, or any other half-way adjective. It is false. It is wrong. It is a horrible, terrible mistake, one that is very difficult to understand. It is further proof that Modern math and physics have followed the same path as Modern art and music and architecture. It can only be explained as a cultural pathology, one where self- proclaimed intellectuals exhibit the most transparent symptoms of rational negligence. They are outlandishly irrational, and do not care that they are. They are proud to be irrational. They believe - due to a misreading of Nietzsche perhaps - that irrationality is a cohort of creativity. Or it is a stand-in, a substitute. A paradox therefore becomes a distinction. A badge of courage. A brave acceptance of Nature's refusal to make sense (as Feynman might have put it.) If we somehow survive this cultural pathology, the future will look upon our time in horror and wonderment. How did we ever reach such fantastic levels of intellectual fakery and denial, especially in a century steeped in the warnings of Freud to beware of just this illness? [Miles Mathis: "Introductory Remarks on Cantor"] http://milesmathis.com/cant.html

§ 062 Matheologians are accustomed to ignore the difference between potential and actual infinity. Usually they even deny that there is any difference. In this paragraph I will make another attempt to show this difference.

Potential infinity is a never ending, never completed sequence or chain of steps like the following, constructed of initial segments of the ordered set of positive even numbers.

|{2}| = 1 < 2 |{2, 4}| = 2 < 4 |{2, 4, 6}| = 3 < 4, 6 |{2, 4, 6, 8}| = 4 < 6, 8 |{2, 4, 6, 8, 10}| = 5 < 6, 8, 10 |{2, 4, 6, 8, 10, 12}| = 6 < 8, 10, 12 ...

This sequence is not more than all its terms. It cannot surpass all its terms. It is incapable of reaching a cardinality larger than every positive even number, the failure getting clearer and clearer with every term at every step.

The existence of alephs, however, requires actual infinity, completed infinity, finished infinity. But then we have to swallow results like the following:

Consider an urn and infinitely many actions performed within one hour (the first one needs 1/2 hour, the second one 1/4 hour, the third one 1/8 hour and so on).

Fill in 1, 2, remove 1 Fill in 3, 4, remove 2 Fill in 5, 6, remove 3 continue.

After one hour the urn is empty, because for every number the time of removal can be determined. So for the set X of numbers residing in the urn we obtain

LimtimeØ1h |X| ∫ |LimtimeØ1h X|

This prevents any application of set theory to reality although it was Cantor's outspoken aim to apply set theory to reality (cp., e.g., his letter to Mittag-Leffler of Sept. 22, 1884). And it prevents any application to mathematics too, because in mathematics the limit of the sequence

21 2.1 432.1 43.21 6543.21 654.321 ... is not 0 but ¶.

Or assume the existence of all rational numbers of [0, 1], enumerated, for instance, according to Cantor's method, starting with 0. If they all exist, then all permutations should exist, because each number is in finite distance from the first number 0, enumerated by 1. But then also the permutation with all rational numbers enumerated and sorted according to their magnitude should be obtainable within aleph_0 steps and, therefore, should exist, shouldn't it? This prevents any application of set theory to mathematics either.

Therefore only potential infinity can prevail. This is the true reason why matheologians deny to recognize the difference between potential and actual infinity. While most of the followers certainly honestly claim that they never thought about that problem, I cannot believe that the leading matheologians always have observed the same ignorance.

§ 063 {{It is impossible to order or to well-order items that cannot be identified. Zermelo's corresponding proof is incorrect - just as is his edition of Cantor's letters:}}

[...] the final section of the correspondence with Cantor starts only in July 1899. This was the part from which extracts were published in the edition of Cantor's papers by Zermelo using the transcriptions made by Cavaillès. [...] The standard of editing of the extracts is bad. [...] The collection begins with a letter from Cantor of 28 July 1899. lt is the most famous of them all [...] lt is often cited in the literature on the foundations of mathematics, and was translated into English in [J. van Heijenoort: "From Frege to Gödel ..." (1967, Cambridge, Mass.) 113-117.] There does not exist a letter in this form. [I. Grattan-Guinness: "The Rediscovery of the Cantor-Dedekind Correspondence", Jahresbericht DMV 76 (1974) p. 126f]

§ 064 Pure mathematics will remain more reliable than most other forms of knowledge, but its claim to a unique status will no longer be sustainable.

For centuries mathematics has been seen as the one area of human endeavor in which it is possible to discover irrefutable, timeless truths. Indeed, theorems proved by Euclid are just as true today as they were when first written down more than 2000 years ago. That the sun will rise tomorrow is less certain than that two plus two will remain equal to four.

However, the 20th century witnessed at least three crises that shook the foundations on which the certainty of mathematics seemed to rest. [Goedel, Four-Color Theorem, Classification of Finite Simple Groups. ...]

A problem that can be formulated in a few sentences has a solution more than ten thousand pages long. The proof has never been written down in its entirety, may never be written down, and as presently envisaged would not be comprehensible to any single individual. The result is important and has been used in a wide variety of other problems in group theory, but it might not be correct.

These three crises could be hinting that the currently dominant Platonic conception of mathematics is inadequate. As Davies remarks:

{{These}} crises may simply be the analogy of realizing that human will never be able to construct buildings a thousand kilometres high and that imagining what such buildings might "really" be like is simply indulging in fantasies.

We are witnessing a profound and irreversible change in mathematics, Davies argues, which will affect decisively its character:

{{Mathematics}} will be seen as the creation of finite human beings, liable to error in the same way as all other activities in which we indulge. Just as in engineering, mathematicians will have to declare their degree of confidence that certain results are reliable, rather than being able to declare flatly that the proofs are correct.

Davies's article "Whither Mathematics?" (PDF, 448 KB) is available at Mathematics: The Loss of Certainty.

[Science Blog (2005)] http://scienceblog.com/community/older/2005/10/200509095.shtml

§ 065 The induction principle is this: if a property holds for 0, and if whenever it holds for a number n it also holds for n + 1, then the property holds for all numbers. For example, let θ(n) be the property that there exists a number m such that 2ÿm = nÿ(n + 1). Then θ(0) (let m = 0). Suppose 2ÿm = nÿ(n + 1). Then 2ÿ(m + n + 1) = (n + 1)ÿ((n + 1) + 1), and thus if θ(n) then θ(n + 1). The induction principle allows us to conclude θ(n) for all numbers n. As a second example, let π(n) be the property that there exists a non-zero number m that is divisible by all numbers from 1 to n. Then π(0) (let m = 1). Suppose m is a non-zero number that is divisible by all numbers from 1 to n. Then mÿ(n + 1) is a non-zero number that is divisible by all numbers from 1 to n + 1, and thus if π(n) then π(n + 1). The induction principle would allow us to conclude π(n) for all numbers n. The reason for mistrusting the induction principle is that it involves an impredicative concept of number. lt is not correct to argue that induction only involves the numbers from 0 to n; the property of n being established may be a formula with bound variables that are thought of as ranging over all numbers. That is, the induction principle assumes that the natural number system is given. A number is conceived to be an object satisfying every inductive formula; for a particular inductive formula, therefore, the bound variables are conceived to range over objects satisfying every inductive formula, including the one in question. In the first example, at least one can say in advance how big is the number m whose existence is asserted by θ(n): it is no bigger than nÿ(n + 1). This induction is bounded, and one can hope that a predicative treatment of numbers can be constructed that yields the result θ(n). In the second example, the number m whose existence is asserted by π(n) cannot be bounded in terms of the data of the problem. lt appears to be universally taken for granted by mathematicians, whatever their views on foundational questions may be, that the impredicativity inherent in the induction principle is harmless - that there is a concept of number given in advance of all mathematical constructions, that discourse within the domain of numbers is meaningful. But numbers are symbolic constructions; a construction does not exist until it is made; when something new is made, it is something new and not a selection from a pre-existing collection. There is no map of the world because the world is coming into being. [Edward Nelson: "Predicative arithmetic", Princeton University Press, Princeton (1986)] http://www.math.princeton.edu/~nelson/books/pa.pdf http://www.hs-augsburg.de/~mueckenh/GU/GU11.PPT#417,66,Folie 66

§ 066 [...] according to our view it is meaningless to talk about the set of all points of the continuum.

Numbers facilitate counting, measuring, and calculating.

The numbers belong to the realm of thinking, the continuum belongs to the realm of visualizing. I repeat, for me the continuum is not identical with the set —.

[...] the logical short circuit results from transforming the infinitely increasing number of digits of the potential infinite into an actual infinite and identifying ◊2 with the never ending decimal representation 1.4142 ...

[Detlef Laugwitz: "Zahlen und Kontinuum", BI, Zürich (1986)]

Or the short circuit results from identifying 2-1 + 2-2 + ... + 2-k + ... with the natural number 1.

Limnض 1/n = 0 means the variable number 1/n will never be 0. We'll have always have 1/n > 0. In modern language: " n: 1/n > 0.

Similarly, -1 -2 -n Limnض 2 + 2 + ... + 2 = 1 does not mean anything else than 2-1 + 2-2 + ... + 2-n = 1 - 2-n and 2-n is never 0.

No last natural number L can be defined. And even if it could, 2-L = 0 would certainly not be satisfied. Therefore 2-1 + 2-2 + ... + 2-n + ... = 1 is only a short cut. Every interpretation as a sum of actually infinitely many terms would be just the short circuit that Laugwitz criticizes.

If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined. [Niels Henrik Abel quoted in G. F. Simmons: "Calculus Gems", New York (1992)]

With the exception of the geometric series, there does not exist in all of mathematics a single infinite series whose sum has been determined rigorously. [Niels Henrik Abel quoted in E. Maor: "To infinity and beyond", Birkhäuser, Boston (1986)]

I don't know which of the two quotes is more to the point. But that does not matter. In its important clause they are sufficiently concurrent: "... there is in all of mathematics not a single infinite series whose sum has been rigorously determined" because the finished infinite is not part of mathematics and unfinished series haven't fixed sums.

§ 067 Cantor’s Problem

What is born on Sacred Cows is as much religion as is any theology.

This paper offers a contrary conclusion to Cantor’s argument, together with implications to the theory of computation. [...] We first dispose of misconceptions arising of terms. The word "infinite" is a fabrication of mental ether - a vacuous expression lacking mathematical precision - with many mystical connotations. [...] We define a set to be computationally-countable if once initiated, producing a monotonic sequence of integers, the process of counting - whatever that might be - at some future time is assured to cease [...]. We define a set to be computationally- uncountable where once initiated, exclusive association being constant, the process of counting does not cease. [...] We will use the term “denumerable” where we mean to infer the more conventional concept of “countable” meaning an injection or bijection with the set of integers. [...] The principal set [...] S of all strings composed of elements from the set containing the symbols 0 (zero) and 1 (one). How many elements are in the set S? [...] we can then repeat the concatenation process . Independent of whatever the cardinality of S was presumed to be when we began, this process will "count" our discovery of additional elements without ceasing. Such would seem to reasonably assure the set S is computationally-uncountable; regardless any initial presumptions as to its cardinality. [...] S is denumerable by definition. [...] Assume some set exists for which no injection into S exists. Let that set be represented by the symbol Q. We first partition all plausible sets into two disjoint subsets: Q1 and Q2. Let Q1 be those sets for which representation of elements is derived from a computationally-countable set of symbols. Let Q2 be those sets for which representation of elements is derived from a computationally-uncountable set of symbols. Let Q’ be some set in Q1 defined above with representation of elements of Q’ derived of strings of symbols in some computationally- countable set P’. The symbols in P’ can be placed into a 1 - 1 correspondence with some set of integers in Ÿ, thus in 1 - 1 correspondence with some set of elements of S. Then by simple rewriting of their representation all elements of Q’ are mapped to elements of S. The consequence of said mapping is at least an injection from Q’ to a set contained in S. It follows then that for any set Q’ that |Q’| is not greater than |S|. Let Q’’ be some set in Q2 defined above with representation of elements of Q’’ derived of strings of symbols in some computationally- uncountable set P’’. By applying Cantor’s Diagonalization method, having established for any subset of P’’ an injective assignment from S one can obtain yet another element in S to assign the next yet unassigned element in P’’, using the assignments thus far made to assure the mapping is a proper injection (i.e. 1 - 1). The injection of the set of symbols into S, while not computable in total, arguably exists. It then follows of the same principles as held for Q’ above that S contains at least one set into which an injection from Q’’ is formed through simple substitution rewriting of the symbols of each string. It can therefore be concluded that there exists no set Q having cardinality greater than the cardinality of S, as any set Q can be shown to be either in Q1 or Q2 as defined above, and thus “contained” by correspondence with a set contained in S through an injection into S. The later is a result of considered importance to the conclusions that Cantor’s Problem does admit. Now to the essential question of whether — has an injection into any set contained in S. Since — is represented by strings over the set P’ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, +, ‐, .}, a computationally- countable (finite) set, it can be concluded that S contains at least one set into which an injection from — exists because — is in Q1 as later was defined above. Such a conclusion contradicts Cantor's conclusion as to the relative cardinality of — and Ÿ; given that it has been shown that Ÿ has a bijection to S by and — has at least an injection into S. If we assume that — exists in a bijection relation to S, such grants at best |—| = |Ÿ|. [...] set C = {c | c {m, w}*} [...] is contained in Q1, as defined above. By direct substitution of symbols m Ø 0 and w Ø 1, any string in C can be rewritten to a string in S. Given that there does not exist in C two strings c1, c2 such that c1 = c2, the rewriting of strings in C produces by definition a bijection with S. This results in a contradiction for two reasons. The bijection with S is such that C is denumerable by the same means with which S is placed into bijection relation with Ÿ. While at the same time, S is not denumerable if the Diagonalization argument is valid. The latter then stands in contradiction of the bijection between Ÿ and S. [Charles Sauerbier (2009)] http://arxiv.org/ftp/arxiv/papers/0912/0912.0228.pdf

§ 068 To most mathematicians, the title of this article will, I suppose, appear a bit strange: it is so obvious that 265536 is a natural number that there would seem to be no rational basis for questioning it. Yet there have been objections to the claim that all such exponential expressions name a natural number, two of the best known being due to Paul Bernays and Edward Nelson. Bernays, in "On Platonism in Mathematics", rhetorically questions whether 67257729 can be represented by an " numeral" (he does not, however, press the discussion). By contrast, Nelson, in "Predicative Arithmetic", develops a large body of theory which he then advances to support his belief that 265536 is not a natural number or that, more generally, exponentiation is not a total function. [...] For while it does not limit the use of induction it does imply that the effect of induction is context-dependent. It also implies that when the objects of discussion are linguistic entities (and in this paper the position advocated is that "natural numbers" or better "natural number notations" are linguistic entities) then that collection of entities may vary as a result of discussion about them. A consequence of this is that the "natural numbers" of today are not the same as the "natural numbers" of yesterday. Although the possibility of such denotational shifts remains inconceivable to most mathematicians, it seems to be more compatible with the history both of the cultural growth (and of growth in individuals) of the number concept than is the traditional, Platonic picture of an unchanging, timeless, and notation-independent sequence of numbers. [David Isles: "What evidence is there that 265536 is a natural number?", Notre Dame Journal of Formal Logic, 33,4 (1992) 485-480.] http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.ndjfl/1093 634481&page=record

§ 069 Summary: A study of the philosophical and historical foundations of infinity highlights the problematic development of infinity. Aristotle distinguished between potential and actual infinity, but rejected the latter. lndeed, the interpretation of actual infinity leads to contradictions as seen in the paradoxes of Zeno. lt is difficult for a human being to understand actual infinity {{not only for a human being but for every intelligent being that can understand the intended meaning of actual infinity at all}}. Our logical schemes are adapted to finite objects and events. {{Praises be!}} Research shows that students focus primarily on infinity as a dynamic or neverending process {{they may have read Fraenkel and Gödel, cp. KB 090923}}. Individuals may have contradictory intuitive thoughts at different times without being aware of cognitive conflict. The intuitive thoughts of students about both the actual (at once) infinite and potential (successive) infinity are very complex. The problematic nature of actual infinity and the contradictory intuitive cognition should be the starting point in the teaching of the concept infinity.

Dubinsky (2005b:261) wys ook daarop dat studente glo dat die vergelyking 0,99999... = 1 vals is. {{Taken literally, they are right. Since the actual infinite does not exist, 0.99999... is at best by definition to be interpreted as 1 - as a limit without explicitly written limit symbol.}} Studente dink dat (1) 0,99999... 'n bietjie kleiner is as 1, die naaste wat jy daaraan kan kom sonder om dit werklik te bereik; en (2) die verskil tussen die twee (09999... en 1) oneindig klein is {{potentially, yes}} - Tall noem dit die intreding van die ifinitesimale (1978:6); of (3) dat 0,9999... die laaste getal voor 1 is. [p. 76]

This thesis contains beautiful and several grahics by Escher.

[Rinette Mathlener: "Die problematiek van die begrip oneindigheid", Dissertation, Magister Educationis, Universiteit van Suid-Afrika(2008)] http://uir.unisa.ac.za/handle/10500/1927 http://uir.unisa.ac.za/bitstream/handle/10500/1927/dissertation.pdf?sequence=1

§ 070 The iterative conception can only help resolve the paradoxes if we view it as clarifying the intuitive concept of a collection, not as introducing a new, distinct “set” concept. But this clarification is elusive, as can be seen by looking at some of the versions of the conception which have appeared in the literature:

Sets are "formed", "constructed", or "collected" from their elements in a succession of stages ... ([18], p. 506)

According to the iterative conception, sets are created stage-by-stage, using as their elements only those which have been created at earlier stages. ([19], p. 183)

In the metaphor of the iterative conception, the steps that build up sets are “operations” of “gathering together” sets to form “new” sets. ([21], p. 637)

Thus a set is formed by selecting certain objects ... we want to consider a set as an object and thus to allow it to be a member of another set ... When we are forming a set z by choosing its members, we do not yet have the object z, and hence cannot use it as a member of z. ([22], p. 322-323)

It should be apparent from this selection that the nature of the set-forming operation is extremely unclear. There seems to be no general agreement even as to whether this is an actual operation which could in any sense be carried out, or instead some kind of impenetrable metaphor. The problem is apparent in Boolos’s remark that “a rough statement of the idea ... contains such expressions as ‘stage’, ‘is formed at’, ‘earlier than’, ‘keep on going’, which must be exorcised from any formal theory of sets. From the rough description it sounds as if sets were continually being created, which is not the case” ([4], p. 491). Yet Boolos does not follow his rough statement with a more informative informal description that avoids the objectionable phrases, and it seems doubtful that he could. Without these expressions there is no informal description. This difficulty is connected to the ontological problem, about which none of the authors cited above has anything meaningful to say: if we have no idea what sets are supposed to be,obviously there is little we can say about how they are supposed to be formed. Yet the idea that sets are in some sense “formed” from elements which enjoy some kind of “prior” existence is crucial to the iterative conception’s ability to evade the classical paradoxes. The point is supposed to be that the set of all sets, or the set of all ordinals, or Russell’s set, are illegitimate on the iterative account precisely because they cannot be “formed”. So it seems fair to say that the iterative conception successfully deals with the paradoxes only to the extent that it presents a clear picture of the operation of set formation, which is to say, not at all.

[2] P. Benecerraf and H. Putnam, eds., Philosophy of Mathematics: Selected Readings (second edition), 1983. [4] G. Boolos, The iterative conception of set, in [2], 486-502. [9] K. Gödel, What is Cantor’s continuum problem?, in [2], 470-485. [18] C. Parsons, What is the iterative conception of set?, in [2], 503-529. [19] M. D. Potter, Iterative set theory, Philosophical Quarterly 43 (1993), 178-193. [20] W. V. Quine, Immanence and validity, Dialectica 45 (1991), 219-230. [21] M. F. Sharlow, Proper classes via the iterative conception of set, J. Symbolic Logic 52 (1987), 636-650. [22] J. R. Shoenfield, Axioms of set theory, in Handbook of Mathematical Logic, J. Barwise, ed., 1977, 321-344.

[Nik Weaver: "Is set theory indispensable?" (2009)] http://arxiv.org/pdf/0905.1680.pdf

§ 071 The Hausdorff Sphere Paradox [...] (here X, Y, Z are disjoint sets which nearly cover the sphere, and X is congruent to Y, in the sense that a rotation of the sphere makes X coincide with Y, and likewise Y is congruent to Z. But what is extraordinary is the claim that X is also congruent to the union of Y and Z, even though Y ∫ Z). We are, like Poincaré and Weyl, puzzled by how mathematicians can accept and publish such results; why do they not see in this a blatant contradiction which invalidates the reasoning they are using? Nevertheless, L. J. Savage (1962) accepted this as literal fact and, applying it to probability theory, said that someone may be so rash as to blurt out that he considers congruent sets on the sphere equally probable; but the Hausdorff result shows that his beliefs cannot actually have that property. The present writer, pondering this, has been forced to the opposite conclusion: my belief in the existence of a state of knowledge which considers congruent sets on a sphere equally probable, is vastly stronger than my belief in the soundness of the reasoning which led to the Hausdorff result. Presumably, the Hausdorff sphere paradox and the Russell Barber paradox have similar ; one is trying to define weird sets with self-contradictory properties, so of course, from that mess it will be possible to deduce any absurd proposition we please. [...] Russell's theory of types can dispose of a few paradoxes, but far from all of them. Even with the best of good will on both sides, it would require at least another generation to bring about the reconciliation of pure mathematics and science. For now, it is the responsibility of those who specialize in infinite set theory to put their own house in order before trying to export their product to other fields. Until this is accomplished, those of us who work in probability theory or any other area of applied mathematics have a right to demand that this disease, for which we are not responsible, be quarantined and kept out of our field. In this view, too, we are not alone; and indeed have the support of many non-Bourbakist mathematicians. [E. T. Jaynes: "Probability Theory: The Logic of Science", (Fragmentary Edition of March 1996)] http://www-biba.inrialpes.fr/Jaynes/cappb8.pdf

§ 072 No matter how much the content of mathematics exploits paradox, mathematicians express dedication to policing their doctrine against inconsistency. Mathematicians do not welcome those who attempt inconsistency proofs of favored theories.

What was profound was that results were contextualized so that they ceased to be inconsistency threats. The Löwenheim-Skolem paradox; Skolem's w-inconsistent model of Peano arithmetic (also the conjunction of Gödel's completeness and incompleteness theorems); w-inconsistency of Quine's "" set theory; independence of the Axiom of Choice; etc. But mathematics had always proceeded like this: e.g. Dedekind had taken Galileo's paradox as the definition of infinity.

The alternative would be that the content of academic mathematics eventuates from sophistry and majority preference. As to the latter, Paul Lorenzen says: You will become famous if you please famous people - and all famous mathematicians like axiomatic set theory.

I will refer to such considerations as professional discipline (or even coercion). If this is the situation, then the profession will protect itself from heresies and criticism not by superior reasoning {{no doubt!}}, but by additional professional discipline in conjunction with additional casuistry.

At the beginning of rational mathematics stands the result concerning the incommensurability of the side and diagonal of a square. This severe embarrassment, which can be conceived as an elementary refutation of mathematics, was instead co-opted as a program for the development of mathematics. (With legitimacy of proof by contradiction as a crucial consideration allowing this.) And yet the status of irrational numbers and the continuum of reals has been questionable to this very day.

Already a precedent for this approach has been provided by historical studies of the Axiom of Choice (with the Banach-Tarski paradox) by Gregory Moore and Stan Wagon. From their excavations, we learn that Émile Borel published a book, Les Paradoxes de l'Infini (3rd ed., Paris, 1946), which on p. 210 said that the Banach-Tarski paradox amounts to an inconsistency proof of the Axiom of Choice.

I do not take a compliant view of mathematics. I consider it as a historically given doctrinal institution; placing in suspension any claim that there is an ideal perfect mathematics. Given Hennix's combined historical and rational scrutiny of the development of mathematical doctrine, I will propose that the main factor in the establishment of "truth" in mathematics is professional procedure and discipline.

There already are many results which are inconsistency proofs in effect. Only professional discipline forestalls the obvious interpretation of these proofs as inconsistency proofs. Quasi- inconsistency proofs are neutralized or co-opted by negotiation and the addition of layers of interpretation.

The most celebrated results of the twentieth century (probably earlier centuries as well) came from skirting paradox while claiming not to land in it. A paradoxical positing is made deliberately, and then is deflected so that, as interpreted, it is not a contradiction.

Truth is negotiated on the basis of manipulation of import by distorting interpretations. Interpretation takes the form of discarding traditional concerning mathematical structure: the privileged position of Euclidian geometry; the invariance of dimension; the association between integer and magnitude; uniqueness of the natural number series; etc.

From time to time, results are discovered which patently embarrass the conventional wisdom, or controvert popular tenets. [The Gödel theorems.] Then follows a political manipulation, to distort the unwanted result by interpretation so that it is seen to "enhance" the popular tenet rather than to controvert it.

Even if my sense of the situation is right, the appearance of such a professionally compelling proof would be a more a matter of packaging and selling than anything else. [...] The biggest hurdle such an attempted proof faces is professional discipline. Whether inconsistency proofs are recognized to have occurred is subject to entirely "political" manipulation.

So, even though, for example, the Hausdorff-Banach-Tarski paradox has been called the most paradoxical result of the twentieth century, classical mathematicians have to convince themselves that it is natural, because it is a consequence of the Axiom of Choice, which classical mathematicians are determined to uphold, because the Axiom of Choice is required for important theorems which classical mathematicians regard as intuitively natural.

[Henry Flynt: "Is mathematics a scientific discipline?" (1994)] http://www.henryflynt.org/studies_sci/mathsci.html

§ 073 Consistency and Madness

Go to a mental hospital and I'll bet you will meet people there who call themselves Napoleon, or Jesus Christ, or Elvis Presley, or . For the sake of simplicity, let us take the man with the Einstein complex. Now suppose that you take such a man apart and you decide to talk with him, in order to convince him that Albert Einstein is dead and buried. And that his real name is Johnson. And that he is just the man around the corner. No genius at all. Do you think you are going to be successful? You talk to him for more than one hour, trying to convince him that he should give up his picture of the world. At last, you ask him if he has understood your arguments. I'll bet his answer will be like the following: "Yes, of course I understand ! Because Albert Einstein, he is a genius. He is so clever that he can understand, of course, any of your arguments. That's why. And what's more, I am Albert Einstein." Now replace "Einstein" by any mainstream mathematician and you're done. Nobody can deny that people in a mental hospital have a consistent picture of the world. Now the good news of consistency is that, once you are on the right track, you will remain on the right track. The axiom system for Euclidian Geometry is a good example of this manner of being consistent. But here comes the bad news. Once you are on the wrong track, you will always be on the wrong track. There is no way to tell a mathematician that the axioms of the Zermelo Fraenkel / axiom of Choice (ZFC) system are "not good". Because then he will defend himself as our would-be Einstein character did. All of his arguments will be consistent with the system he believes in. He wants to remain in his vicious circle, safe and well. Calling everybody else a "dude" and a "crank" and a "zealot". And there is no way out. There is no cure for his madness. Because mathematics is what mathematicians do. [Han de Bruijn: Natural philosophy] http://www.alternatievewiskunde.nl/QED/natural.htm#oo

§ 074 Dedekind tried to describe an infinite class by saying that it is a class which is similar to a proper subclass of itself. [...] I am to investigate in a particular case whether a class is finite or not, whether a certain row of trees, say, is finite or infinite. So, in accordance with the definition, I take a subclass of the row of trees and investigate whether it is similar (i.e. can be co-ordinated one-to-one) to the whole class! (Here already the whole thing has become laughable.) It hasn’t any meaning; for, if I take a "finite class" as a subclass, the attempt to co-ordinate it with the whole class must eo ipso fail: and if I make the attempt with an infinite class - but already that is a piece of nonsense, for if it is infinite, I cannot make an attempt to co-ordinate it. [...] An infinite class is not a class which contains more members than a finite one, in the ordinary sense of the word "more". If we say that an infinite number is greater than a finite one, that doesn't make the two comparable, because in that statement the word "greater" hasn’t the same meaning as it has say in the proposition 5 > 4!

The form of expression "m = 2n correlates a class with one of its proper subclasses" uses a misleading analogy to clothe a trivial sense in a paradoxical form. (And instead of being ashamed of this paradoxical form as something ridiculous, people plume themselves on a victory over all prejudices of the understanding). It is exactly as if one changed the rules of chess and said it had been shown that chess could also be played quite differently.

When "all apples" are spoken of, it isn’t, so to speak, any concern of logic how many apples there are. With numbers it is different; logic is responsible for each and every one of them.

Mathematics consists entirely of calculations.

In mathematics description and object are equivalent. {{Therefore numbers that cannot be described cannot exist.}} "The fifth number of the number series has these properties" says the same as "5 has these properties". The properties of a house do not follow from its position in a row of houses; but the properties of a number are the properties of a position.

[L. Wittgenstein: "Philosophical Grammar", Basil Blackwell, Oxford (1969), quoted from E.D. Buckner: "THE LOGIC MUSEUM" (2005), unfortunately no longer on the web.] http://www.amazon.de/Philosophical-Grammar-Ludwig- Wittgenstein/dp/0631123504/ref=sr_1_1?ie=UTF8&qid=1286647502&sr=1-1

There is no path to infinity, not even an endless one. [§ 123]

It isn't just impossible "for us men" to run through the natural numbers one by one; it's impossible, it means nothing. […] you can’t talk about all numbers, because there's no such thing as all numbers. [§ 124]

An "infinitely complicated law" means no law at all. [§ 125]

There's no such thing as "all numbers" simply because there are infinitely many. [§ 126]

Does the relation m = 2n correlate the class of all numbers with one of its subclasses? No. It correlates any arbitrary number with another, and in that way we arrive at infinitely many pairs of classes, of which one is correlated with the other, but which are never related as class and subclass. Neither is this infinite process itself in some sense or other such a pair of classes.

In the superstition that m = 2n correlates a class with its subclass, we merely have yet another case of ambiguous grammar. [§ 141]

Generality in mathematics is a direction, an arrow pointing along the series generated by an operation. And you can even say that the arrow points to infinity; but does that mean that there is something - infinity - at which it points, as at a thing? Construed in that way, it must of course lead to endless nonsense. [§ 142]

If I were to say "If we were acquainted with an infinite extension, then it would be all right to talk of an actual infinite", that would really be like saying, "If there were a sense of abracadabra then it would be all right to talk about abracadabraic sense perception". [§ 144]

Set theory is wrong because it apparently presupposes a symbolism which doesn't exist instead of one that does exist (is alone possible). It builds on a fictitious symbolism, therefore on nonsense. [§ 174]

[L. Wittgenstein: "Philosophical Remarks", quoted from E.D. Buckner: "THE LOGIC MUSEUM" (2005), unfortunately no longer on the web]

§ 075 Several examples are used to illustrate how we deal cavalierly with infinities and unphysical systems in physics. [...] At this point, we can ask what all this {{Cantor's infinites}} implies for physics and physical systems. There are significant implications for physical systems if we accept the major that rather than being merely potential, real infinities do exist. We shall now cite some examples to illustrate some of these implications. The case of an infinitely long line charge distribution is a simple example. In this case both the length and the electric charge are of cardinality C for the continuum and consequently the ratio λ representing the charge density is finite. The case of the thermodynamic limit invoked in statistical mechanics is a fascinating counter-example. If we treat them as real infinities, then the ratio N/V involves the number of atoms which is countable and thus of cardinality ¡0 whereas the volume of the system is of cardinality C x C x C = C. For this ratio to be finite we must require that C must equal a finite number times ¡0. This last statement is however, not true for real infinities. Hence it follows that the finiteness of the number density N/V would not have been possible if N and V were real infinities. Indeed the finiteness of the number density follows from the fact that each of these quantities is finite to begin with, as was discussed earlier. [...] We conclude by making a few general observations. Thermodynamic systems are devoid of infinities and are inherently finite. N is countable and large and V is measurable and macroscopically large but all physical parameters are finite and measurable and finite, including the number density. There is a class of physical systems containing infinities but which can be re-examined by using methods which have successfully prevailed in thermodynamics and statistical mechanics, with a view to resolve the problem infinities in these systems. [...] Finally there may be physical systems containing real infinities which cannot be transformed away. These systems may perhaps be understood only by a re-examination based on Cantor’s transfinite mathematics. {{May or may not.}} In this context, it is useful to remember that Cantor’s theory is well grounded in physical reality {{where can we find that reality?}}: it is based on arithmetic and set theory. {{That holds also for astrology, , and Traumdeutung. But an argument to join one of these sects cannot be obtained from this paper.}}

{{Footnote 9}} An interesting point in the history of mathematics is the continuum hypothesis: that there is no cardinality bigger than ¡0 and smaller than C. This notion is neither proved nor disproved. {{Meanwhile it has been proved: there is no cardinality bigger than ¡0. See, for instance, § 062}}

[P. Narayana Swamy: "Infinities in Physics and Transfinite numbers in Mathematics" (1999)] http://arxiv.org/abs/math-ph/9909033

§ 076 Mathematics, we hold, deals with multiple models of the world. It is not subsumed in any one big model or by any one grand system of axioms. The idea that set theory is relative is not new; it was clearly stated for axiomatic set theory by Skolem in 1922 [9]. We intend simply to draw some of the philosophic consequences of that relativity. For the Platonist, there is a real world of sets, existing forever, described only approximately by the Zermelo-Fraenkel axioms or by their modifications. It may be that some final insight will give a definite axiom system, but the sets themselves are the underlying mathematical reality. In our view, such a Platonic world is speculative. It cannot be clearly explained as a matter of fact (ontologically) or as an object of human knowledge (epistemologically). Moreover, such ideal worlds rapidly become too elaborate; they must display not only the sets but all the other separate structures which mathematicians have described or will discover. The real nature of these structures does not lie in their often artificial construction from set theory, but in their relation to simple mathematical ideas or to basic human activities. Hence, we hold that mathematics is not the study of intangible Platonic worlds, but of tangible formal systems which have arisen from real human activities.

Many students of set theory do not follow what I have called the "Grand Set Theoretic Foundation" but instead follow Cantor to emphasize the intuitive notion of a set as a collection which is a real object in its own right. For them set theory is not subsumed by the Zermelo- Fraenkel axiom system or by any other first-order formal system. It may be studied formally by other means; using infinitary languages or second-order logic Such Cantorian sets are just as real as numbers. Indeed, one might say that number theory is formalized only in part by Peano's arithmetic in just the way set theory is formalized, but only in part, by Zermelo-Fraenkel. (There can be true properties of whole numbers not demonstrated in Peano arithmetic.) From this point of view, set theory is just another branch of mathematics {{if at all}}. If in this view set theory is not taken to be the foundation of mathematics {{relieving}}, it can be assimilated with our proposal that mathematics consists of formal disciplines derived from a variety of human activities. {{Human activities? Is omega-tasking a human activity? Well, in some sense. At least believing in is - like worshipping of the devil (with a lot of formal rules).}}

The various earlier of mathematics [...] each arose out of the dominant aspects of mathematics as then understood. For example, Platonism arose in Greece and applied to mathematics there because it fitted Greek geometry; it has been popular among mathematicians recently because it fitted well with the view that mathematics derives from axioms for sets. arose together with the discovery and formalization of mathematical logic. Intuitionism was the child of emphasis on numbers as the starting point of mathematics and on intuition as a basis of topology. Formalism arose with the development of axiomatic methods and the discovery that proof theory might give consistency proofs for abstract mathematics. sprang from the 19th-century view of mathematics as almost coterminal with theoretical physics; it was much influenced by Kant's dichotomy between analytic and synthetic. Now we search for a philosophy of mathematics better attuned to the present state of the subject.

2. Paul C. Eklof, Whitehead's problem is undecidable, this Monthly, 83 (1976) 775-787. 9. Th. Skolem, Einige Bemerkungen zur axiomatische Begründung der Mengenlehre. Fifth Congress of Scandinavian Mathematicians, 1922, Helsingfors, 1923, pp. 2 17-232.

[Saunders Mac Lane: "Mathematical models: A sketch for the philosophy of mathematics", The American Mathematical Monthly, Vol. 88, No. 7 (1981) 462-472.] http://home.dei.polimi.it/schiaffo/TFIS/philofmaths.pdf

The answer is: MatheRealism. mhtml:http://www.hs-augsburg.de/~mueckenh/MR.mht http://arxiv.org/abs/math/0505649

§ 077 Well known is the story of Tristram Shandy [Laurence Sterne: "The Life and Opinions of Tristram Shandy" (1759-1767)] http://en.wikipedia.org/wiki/The_Life_and_Opinions_of_Tristram_Shandy,_Gentleman who undertakes to write his biography, in fact so pedantically, that he needs for each of the first days of his life a full year. Of course he will never get ready if continuing that way. But if he would live infinitely long (for instance a countable infinity of years), then his biography would get "ready", because every day in his life, how late ever, finally would get its description. [Adolf A. Fraenkel: "Einleitung in die Mengenlehre" 3rd ed., Springer, Berlin (1928) p. 24] Original German Text: http://www.hs-augsburg.de/~mueckenh/GU/GU12c.PPT#395,21,Folie 21

... for instance the story of Tristram Shandy who writes his autobiography so pedantically that the description of each day takes him a year. If he is mortal he can never terminate; but if he lived forever then no part of his biography would remain unwritten, for to each day of his life a year devoted to that day's description would correspond. [A.A. Fraenkel, A. Levy: "Abstract Set Theory", North Holland, Amsterdam (1976) p. 30]

§ 078 Digital Philosophy (DP) is a new way of thinking about the fundamental workings of processes in nature. DP is an atomic theory carried to a logical extreme where all quantities in nature are finite and discrete. This means that, theoretically, any quantity can be represented exactly by an integer. Further, DP implies that nature harbors no infinities, , continuities, or locally determined random variables. This paper explores Digital Philosophy by examining the consequences of these premises. [...] Digital Philosophy makes sense with regard to any system if the following assumptions are true: All the fundamental quantities that represent the state information of the system are ultimately discrete. In principle, an integer can always be an exact representation of every such quantity. For example, there is always an integral number of neutrons in a particular atom. Therefore, configurations of bits, like the binary digits in a computer, can correspond exactly to the most microscopic representation of that kind of state information. In principle, the temporal evolution of the state information (numbers and kinds of particles) of such a system can be exactly modeled by a digital informational process similar to what goes on in a computer. Such models are straightforward in the case where we are keeping track only of the numbers and kinds of particles. For example, if an oracle announces that a neutron decayed into a proton, an electron, and a neutrino, it’s easy to see how a computer could exactly keep track of the changes to the numbers and kinds of particles in the system. Subtract 1 from the number of neutrons, and add 1 to each of the numbers of protons, electrons, and neutrinos. The possibility that DP may apply to various fields of science {{except transfinite set theory}} motivates this study. http://www.digitalphilosophy.org/ http://en.wikipedia.org/wiki/Edward_Fredkin

§ 079 A second negative effect of transfinite theory is that by positioning transfinite theory as the basis arithmetic and logic, transfinite theorists have accomplished the unintended goal of basing mathematics on mystical concepts: namely, the completed infinity. Such a result is highly attractive to the mystical mind. Again, classical logicians would abhor the idea of basing reasoning on mystical concepts, but the mystical mind adores it... gaining further acceptance for transfinite theory. This leads to one of the weirdest twists in modern mathematics: The intuitionalist school, led by LEJ Brouwer actually demanded a higher degree of logical rigor than the Logicists school that was happy with a mathematics founded on the slippery slope of transfinite theory, completed infinities and paradoxes. Of course, we have seen the same twists in politics. The "people's" party is generally a pseudonym for a dictatorship. Tax reform almost always leads to higher taxes. Intuitionism in mathematics does not mean that people rely on their "intuition." Intuitionist school demands that all axioms be based on comprehendible ideas. The logistical schools holds that the axioms don't really matter, so long as symbolic logic used is consistent. The ideas expressed by the symbols can be insane. [...] In my opinion, the fact that schools no longer teach logic is the worst effect of transfinite theory. [Kevin Delaney: "Curing the Disease"] http://descmath.com/diag/cure.html

§ 080 Borel's thesis is that the overwhelming majority of numbers will always remain inaccessible to the human race as we know it, in the sense that it will never be possible to define these numbers effectively in such a manner that any two mathematicians will be certain that they are speaking about one and the same entity.

If an enumerable set, such as that of the natural numbers, is considered instead of the unit interval, then the assignment of equal to the elements of this set reduces to zero the global probability of any number of accessible integers, which, according to Borel, is absurd because it precludes the possibility of ever getting one of these numbers, so that every choice leads to an inaccessible number. {{Think of a number ... The choice is very restricted.}}

Borel discusses the familiar decomposition of the circumference of a circle into an enumerable number of mutually exclusive, congruent sets of points. He asserts that it is not possible to attribute equal probabilities to these sets without running into contradictions, and that it is therefore necessary to attribute unequal probabilities to them. "But then we contradict the Euclidean principle of equality, according to which two superposable figures are equal." As the construction of these sets "requires the use of Zermelo's axiom, our conclusion is that it is necessary to choose between Zermelo's axiom and Euclid's axiom according to which two superposable figures are equal, that is to say, identical from all points of view, and that, in particular, equal probabilities correspond to them. The simultaneous application of the two axioms leads, in fact, to a contradiction." (Borel, needless to say, chooses "Euclid's axiom.") {{He was just a real mathematician.}} This argument is open to objections. First, it is not inconceivable that a nonmeasurable set can be constructed without the intervention of Zermelo's axiom. Secondly, there is another way in which two superposable figures may be "identical from all points of view," without having equal probabilities correspond to them, and that is, by having no probabilities correspond to them {{why then not starting off with a completely meaningless mathematics?}}. Euclid's axiom cannot be interpreted as stating that congruent figures - if the "figures" in question are not the elementary Euclidean ones {{either they are "figures" hence "elementary Euclidian", or no "figures" at all}} - have probabilities and that these probabilities are equal. [...] Complex numbers were once considered meaningless, whereas today some mathematicians consider Zermelo's axiom and its consequences meaningless. {{In fact, one can discern by this question mathematicians from matheologians.}} Borel's notion of accessibility, although of heuristic significance, seems too subjective, temporal, and, by precluding intrinsically the possibility of delimiting the realm of the accessible {{admitedly, for delimiting paradoxes like that of Hausdorff-Banach-Tarski are well suited}}, vague, according to his own standards, to "define in a precise manner a science of the accessible and of the real." {{Quite that he did - contrary to Zermelo.}}

[F. Bagemihl: "Review: É. Borel: 'Les nombres inaccessibles', Gauthier- Villars, Paris (1952)", Bull. Amer. Math. Soc. 59,4 (1953) 406-409] http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/118 3518025

§ 081 Skolem's conclusion was that the notions of set theory are relative to the universe of sets under consideration (Skolem 1923, p. 224): The axiomatic founding of set theory leads to a relativity of set theoretic notions, and this relativity is inseparably connected with every systematic axiomatization.

Later Skolem even strengthened his opinion by what can be regarded the essential motto of so- called "Skolem relativism" (Skolem 1929, p. 48): There is no possibility of introducing something absolutely uncountable, but by a pure dogma.

Because of this relativistic attitude he avoided traditional set theory and became negligent of problems concerned with semantical notions.

Fraenkel was not sure about the correctness of the proof of the Löwenheim-Skolem theorem. von Neumann instantly recognized the importance of the result but he reacted with scepticism about the possibility of overcoming the weakness of axiomatizations they reveal (1925, p. 240).

[Heinz-Dieter Ebbinghaus: "Ernst Zermelo, An Approach to His Life and Work", Springer (2007) 199f]

§ 082 Cantor offered several proofs that there is no one-one correlation between the real numbers and the natural numbers, but only the presumption that there are infinite numbers can turn whatever impossibility there is here into a seeming demonstration that the number of the real numbers is greater than the number of natural numbers.

Cantor defined real numbers in terms of Cauchy sequences of rationals, specifically in terms of equivalence classes of such sequences (Suppes [26], pp 161, 181). But he merely presumed there were such equivalence classes, not realising that Russell's Paradox, amongst other things, required such a presumption to be justified in any particular case. Now it was on the basis of this assumption that Cantor was able to prove that the real number system, unlike the rational number system, was complete, i.e. that every Cauchy sequence of real numbers, unlike every Cauchy sequence of rational numbers, has a limit ([26], p 185). And that leads to the representation of every real number as not just the would-be of finite decimals, but also a limit which is actually reached, so that a real number is identical with a certain infinite decimal ([26], pp 189, 191). The fact that there cannot be such completed decimal expansions therefore shows that the sets Cantor used to define the reals do not exist: the Cauchy sequences of rationals equivalent to a given one do not form a set (c.f. [27], p 92). Now in addition to the above, well-known "proofs" of the non-denumerability of certain sets, in terms of unending decimals, and in terms of the subsets of the natural numbers, Cantor also gave a proof of the non-denumerability of the reals which rested solely on the completeness of the real number system (Dauben [8], p 51, Grattan-Guinness [10], pp 185-6, see also § 6.2). But the Platonically real limit he there presumed to exist is now shown not to exist, which means we remain compelled to see the infinite as an undifferentiated lack of number.

[8] Dauben, J.W., Georg Cantor, Princeton U.P., Princeton, 1990. [10] Grattan-Guinness, I., From the Calculus to Set Theory 1630-1910, Duckworth, London, 1980. [26] Suppes, P., Axiomatic Set Theory, Van Nostrand, Princeton N.J., 1960. [27] Tiles, M., The Philosophy of Set Theory, Blackwell, Oxford, 1989.

[Hartley Slater: "The Uniform Solution of the Paradoxes" (2004)]

§ 083 As a foundation for mathematics, then, set theory is far less firm than what is founded upon it; for common sense in set theory is discredited by the paradoxes. Clearly we must not look to the set-theoretic foundation of mathematics as a way of allaying misgivings regarding the soundness of classical mathematics. Such misgivings are scarce anyway, once such offenses against reason as the infinitesimal have been set right. [...] For the one thing we insist on, as we sort through the various possible plans for passable set theories, is that our set theory be such as to reproduce, in the eventual superstructure, the accepted laws of classical mathematics. This requirement is even useful as a partial guide when in devising a set theory we have to choose among intuitively undecidable alternatives. We may look upon set theory, or its notation, as just a conveniently restricted vocabulary in which to formulate a general axiom system for classical mathematics - let the sets fall where they may. {{In any case mathematics has priority. If set theory is unable to reproduce the simple and unambiguous fact that the Binary Tree cannot contain more recognizable paths than nodes, then set theory is unsuitable as a foundation of recognizable mathematics describing recognizable reality.}} [Willard V. O. Quine: "The ways of paradox and other essays", Harvard University Press (1966) p. 31f] http://books.google.de/books?id=YReOv31gdVIC&source=gbs_navlinks_s

§ 084 When discussing the validity of the Axiom of Choice, the most common argument for not taking it as gospel is the Banach-Tarski paradox. Yet, this never particularly bothered me. The argument against the Axiom of Choice which really hit a chord I first heard at the Olivetti Club, our graduate colloquium. It’s an extension of a basic logic puzzle, so let’s review that one first. 100 prisoners are placed in a line, facing forward so they can see everyone in front of them in line. The warden will place either a black or white hat on each prisoner’s head, and then starting from the back of the line, he will ask each prisoner what the color of his own hat is (ie, he first asks the person who can see all other prisoners). Any prisoner who is correct may go free. Every prisoner can hear everyone else’s guesses and whether or not they were right. If all the prisoners can agree on a strategy beforehand, what is the best strategy? [...] the first guy counts the total number of white hats. If it is odd, he says “white”, and if it is even, he says “black”. Then the guy in front of him can count the number of white hats he can see, and if differs from the parity the first guy counted, he knows his hat is white. But now the next guy knows the parity of white hats the first guy saw, and whether or not the second guy had a white hat, so he can compare it to the white hats he sees, and find out if his own hat is white. This argument repeats, and so everyone except the first guy guesses correctly. Its interesting to notice that a larger number of hat colors poses no problem here. For any set of hat colors, the prisoners can pick an abelian group structure on. Then, the first prisoner guesses the "sum" of all the hat colors he can see. The next guy can then subtract the sum of the hat colors he sees from the hat color the first guy said to find his own hat color. Again, this argument repeats, and so everyone except the first guy gets out. For the case of black and white, the previous argument used black = 0 (mod 2) and white = 1 (mod 2). This is all well and good, but it doesn’t seem to help the countably infinite prisoners in the second puzzle. Since they can’t hear anyone else’s guess, they can’t set up a similar system for passing on information. So what can they do? First, instead of thinking of hat colors, they just turn white into 1 and black into 0 (like above). Then, a possible scenario of hats on their heads is an infinite sequence of 1’s and 0’s. Call two such sequences ‘equivalent’ if they are equal after a finite number of entries. This is an equivalence relation, and so we can talk about equivalence classes of sequences. Next, the prisoners invoke the Axiom of Choice to pick an element in each equivalence class, which they all agree on and memorize. Now, when they are put in line and get a hat, they will be able to see all but a finite part of the sequence, and so they can all tell what equivalence class they are in. Their strategy is then to guess as if they were in the pre-chosen element in that equivalence class. How well does this work? Well, the sequence they are actually in and the representative element they picked with the axiom of choice must be equivalent, so they are the same after a finite number of entries. Therefore, after a finite number of incorrect guesses, each prisoner will miraculously guess his hat color correctly! This solution is also pretty stable, in that most attempts to make the puzzle harder don’t break it. The warden can know their plan and even know their precise choice of representative sequences. If so, he can make sure any arbitrarily large finite number of them are wrong, but he can’t get an infinite number of them. Also, the number of hat colors can be arbitrarily big; the same solution works identically. This last point is pretty trippy. In the two color case, its very reasonable for any prisoner to guess his hat color correctly, and also for arbitrarily large numbers of them to get it right in a row. Effectively, at no finite point in the guessing do the results of the optimal strategy appear to differ from random guessing. However, if there are uncountably many hat colors, then the probability of any prisoner randomly guessing his hat color is 0. One can reasonably expect no prisoners to be correct for random guessing, so when eventually that first prisoner guesses correctly, the warden should be rightly shocked (though not as shocked as he will be when all but a finite number of prisoners guess correctly). I find this solution deeply troubling to the intuitive correctness of the axiom of choice. Sure, this is based primarily on my intuition for finite things and a naive hope that they should extend to infinities. I think particularly troubling is in the uncountably many colors case, where any given prisoner has no chance to guess his hat color correctly, and yet almost all prisoners are correct. [Greg Muller: "The Axiom of Choice is Wrong", September 13, 2007] http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/ After a talk by Mike O’Connor and an older publication here: http://mathforum.org/wagon/fall05/p1035.html

§ 085 People have asked me, "How can you, a nominalist, do work in set theory and logic, which are theories about things you do not believe in?" ... I believe there is value in fairy tales and in the study of fairy tales. [A. Burdman-Feferman, S. Feferman: "Alfred Tarski - Life and Logic", Cambridge Univ. Press (2004) p. 52] http://books.google.de/books?id=wqktlxHo9wkC&printsec=frontcover&dq=%22+life+and+logic% 22&hl=de&ei=YzQwTO7iCJ2SOK6o3MUB&sa=X&oi=book_result&ct=result&resnum=1&ved=0 CCwQ6AEwAA#v=onepage&q&f=false

§ 086 The reaction to Skolem's results was split. For example, Fraenkel was not sure about the correctness of the proof of the Löwenheim-Skolem theorem, and he seems to have had difficulties in analysing the role of logic with sufficient to understand Skolem's paradox [...]

A "War" Against Skolem

What about Zermelo? When faced with the existence of countable models of first-order set theory, his first reaction was not the natural one, namely to check Skolem's proof and evaluate the result - it was immediate rejection {{there nogthing has changed}}. Apparently, the motivation of ensuring "the valuable parts of set theory" which had led his axiomatizations from 1908 and from the Grenzzahlen paper had not only meant allowing the deduction of important set-theoretic facts, but had included the goal of describing adequately the set-theoretic universe with its variety of infinite . Now it was clear that Skolem's system, like perhaps his own, failed in this respect. Moreover, Skolem's method together with the epistemological consequences Skolem had drawn from his results, could mean a real danger for mathematics {{not at all! - quite the opposite}} like that caused by the intuitionists: In his Warsaw notes W3 he had clearly stated that "true mathematics is indispensably based on the assumption of infinite domains," among these domains, for instance, the uncountable continuum of the real numbers. Hence, following Skolem, "already the problem of the power of the continuum loses its true meaning". Henceforth Zermelo's foundational work centred around the aim of overcoming Skolem's relativism and providing a framework in which to treat set theory and mathematics adequately. Baer speaks of a real war Zermelo had started, wishing him "Heil und Sieg und fette Beute", at the same time pleading for peace [...] However, peace was not to come. [...] a vivid impression of Zermelo's uncomprising engagement, at the same time also revealing his worries: It is well known that inconsistent premises can prove anything one wants; however, even the strangest consequences that Skolem and others have drawn from their basic assumption, for instance the relativity of the notion of subset or equicardinality, still seem to be insufficient to raise doubts about a doctrine that, for various people, already won the power of a dogma that is beyond all criticism. [...] His remedy consisted of infinitary languages {{save nonsense by nonsense}}. [...] Skolem had considered such a possibility, too, but had discarded it because of a vicious circle (Skolem 1923, p. 224): In order to get something absolutely uncountable either the axioms themselves would have to be present in an absolutely uncountably infinite number or one would have to have an axiom which could provide an absolutely uncountable set of first-order sentences. However, in all cases this leads to a circular introduction of higher infinities, that means, on an axiomatic basis higher infinities exist only in a relative sense.

[Heinz-Dieter Ebbinghaus, Volker Peckhaus: "Ernst Zermelo: an approach to his life and work", Springer (2007) p. 199 ff] http://www.springer.com/math/history+of+mathematics/book/978-3-540-49551- 2?cm_mmc=Google-_-Book%20Search-_-Springer-_-0

§ 087 The Banach-Tarski Gyroscope is an intricate mechanism believed to have been constructed using the Axiom of Choice. On each complete rotation counterclockwise, the Banach-Tarski Gyroscope doubles in volume while maintaining its shape and density; on rotating clockwise, the volume is halved. When first discovered, fortunately in the midst of interstellar space, the Banach-Tarski Gyroscope was tragically mistaken for an ordinary desk ornament. Subsequently it required a significant portion of the available energy of the contemporary galactic civilization to reverse the rotation before nearby star systems were endangered; fortunately, the Banach-Tarski Gyroscope still obeys lightspeed limitations on rotation rates, and cannot grow rapidly once expanding past planetary size. After the subsequent investigation, the Banach-Tarski Gyroscope was spun clockwise and left spinning. http://news.ycombinator.com/item?id=411727

§ 088 Consider a sequence of indexed natural numbers 11 21, 32 41, 52, 63 ... The sequence of the indices 1 1, 2 1, 2, 3 ... has limit Ù. But the sequence of natural numbers 1 2, 3 4, 5, 6 ... has limit «, because Ù will be exhausted. What is the limit of the indexed numbers? The indexed numbers may be carried by the balls of a supertask which runs as follows: In the nth step fill into a vase n balls, take off the n - 1 balls that were inside before.

§ 089 In a 1924 paper Tarski proved that seven well-known propositions in cardinal arithmetic whose proofs use the axiom of choice are actually equivalent to the axiom. In the same year he published the first systematic development of a theory of finite sets, based on Zermelo's axioms, but with the negation of the and no axiom of choice. Dedekind and Hausdorff had envisaged such a project, but Tarski was the first to realize it completely. {{Set theory without finished infinity? That's like amalgamating set theory with chocolate - brown too, but delicious.}} [...] The eminent French mathematician Denjoy later incorporated substantial portions of the paper into one of his books [...]. Still in the same year, Tarski published with Stefan Banach a paper [1] that quickly became famous. Its main theorem asserts that any two bounded sets with interior points are equivalent by finite decomposition. For example, a sphere can be decomposed into a finite number of pieces that can be reassembled, using rigid (translations, rotations, and reflections), into two spheres, each of which is congruent to the original one. More dramatically, a sphere the size of a pea can be decomposed into a finite number of pieces that can be reassembled to make a sphere as big as the sun. The proof makes essential use of the axiom of choice. The fact that the axiom has such paradoxical consequences was seen by some as evidence that it should not be accepted. {{Émile Borel published a book, Les Paradoxes de l'Infini, which on p. 210 said that the Banach-Tarski paradox amounts to an inconsistency proof of the Axiom of Choice.}} Between 1923 and 1926, Tarski discovered that a number of implications in cardinal arithmetic that had traditionally been proved using the axiom of choice could in fact be proved without it (at the price of using a more complicated argument). He announced these and many other results in a 1926 paper that was jointly written with Adolf Lindenbaum. There one also finds, for instance, the theorem that the generalized continuum hypothesis implies the axiom of choice. A total of 146 theorems are listed in the paper, all of them without proof. {{Why not? If the theorems were conving.}} We sense that in Lindenbaum, Tarski had found a kindred spirit: the results came so fast that they didn't have time to write them up properly. Sierpinski spent some of the difficult years during World War II, when Warsaw University was closed, working out the proofs of the theorems in this paper. Tarski seems to have had a passion for set theory and, in particular, for cardinal arithmetic during this period. A proof of the continuum hypothesis once came to him in a dream, and the "proof" was so good it took him two weeks to find the mistake. {{Unbelievable. It is so simple: There is no continuum.}} [1] Stefan Banach and Alfred Tarski, "Sur la decomposition des ensembles de points en parties respectivement congruentes," Fundamenta Mathematicae 6, 1924, 244-277. [S. Givant: "Unifying Threads in Alfred Tarski's Work", The Mathematical Intelligencer 21,1 (1999) 47-58]

§ 090 The complete set of rational numbers exists and is countable. Let q1, q2, q3, q4, q5, ... be an enumeration. Consider the first two elements |q1, q2| q3, q4, q5, ... and order them by magnitude: |q1', q2'| q3, q4, q5, ... Consider the first three elements |q1, q2, q3| q4, q5, ... and order them by magnitude. |q1'', q2'', q3''| q4, q5, ... Continue such that in the nth step the first n elements are ordered by magnitude. In the limit all rational numbers have been ordered by magnitude (*) - if limits of non- converging sequences have any meaning. Otherwise, the enumeration of any non-converging sequence is meaningless too. Compare the limit of the sequence (an) with an = (1, 2, 3, ..., n).

(*) The set theoretical limit of the sequences of sets ordered by magnitude can be calculated as follows. (Apostrophes are left out, because although q' may differ from q'', every apostrophized q will remain in the set for ever. By construction every ordered set contains at most one q out of order.) th Let Qk = (q1, q2, q3, ... , qk) be the k initial segment of the set: LimSup (Qk) = …n=1...¶ »k=n...¶ Qk = …n=1...¶ ((q1, ... , qn) » (q1, ... , qn, qn+1) U (q1, ... , qn, qn+1, qn+2) U ...) = …n=1...¶ (q1, q2, q3, ... ) = (q1, q2, q3, ... ) LimInf (Qk) = »n=1...¶ …k=n...¶ Qk = »n=1...¶ ((q1, ... , qn) … (q1, ... , qn, qn+1) … (q1, ... , qn, qn+1, qn+2) … ...) = (q1) » (q1, q2) » (q1, q2, q3) » ... » (q1, q2, q3, ... ) = (q1, q2, q3, ... )

§ 091 No set-theoretically definable well-ordering of the continuum can be proved to exist from the Zermelo-Fraenkel axioms together with the axiom of choice and the generalized continuum hypothesis. [S. Feferman: "Some applications of the notions of forcing and generic Sets", Talk at the International Symposium on the Theory of Models, Berkeley (1963)]

It is well known, that no well-ordering of the reals can be accomplished by mortal humans. But usually matheology can at least prove that things, that cannot be done, can be done. Now even this proof fails! What a pity! On the other hand, well-ordering of the reals must be done. At all costs! Otherwise the hierarchy of infinities would break down and research on inaccessible cardinals would appear like nonsense. So let us pray to the Gods of matheology that they do what no mortal human can do: Well-ordering the real numbers. Perhaps they can even provide a list of all real numbers? But that must be kept secret! Because otherwise the research on inaccessible cardinals ...

§ 092 My second best proof contradicting set theory

1) Define a sequence of points pn in the unit interval: pn = 1/n. These points define intervals Ak = [1/n, 1/(n+1)] for odd n and Bj = [1/n, 1/(n+1)] for even n. The intervals of sort A === and B --- are alternating. If the points are denoted by n, we have something like the following configuration. ...7--6==5--4====3------2======1

Theorem. If two neighbouring points pn and pn+1 are exchanged, the number of intervals remains the same. ...7--6==5--3====4------2======1 The intervals remain alternating. In particular, the number of intervals cannot increase.

2) Define a set of intervals Im in the unit interval such that interval Im has length -m |Im| = 10 and covers the rational number qm of a suitable enumeration of all rational numbers of the unit interval. Then the union of all Im has measure § 1/9. The remaining part of the unit interval has measure ¥ 8/9 and is split into uncountably many singletons. A sketch of the intervals Im ~~~ is given here: ...a b~~~~~~~~~c~~~~~~~~~d~~~~~~e f~~~g~~~h i~~~j k~~~l We cannot exclude intervals within intervals like c~~~d within b~~~e or, alternatively, overlapping intervals like b~~~d and c~~~~e and also adjacent intervals like f~~~g and g~~~h.

3) Let the endpoints pn of the configuration described in (1) move in an arbitrary way, say powered by little ants or by the Gods of matheology. Then it cannot be excluded that the pn and the endpoints of the Im of (2) will coincide (no particular order is required). ...a b~~~~~~~~~c~~~~~~~~~d~~~~~~e f~~~g~~~h i~~~j k~~~l ...3=7------11======5------12=4---2===9-8==10-6==1

As our theorem shows, there will be not more than ¡0 intervals in the end position. This includes the set of Im and the set of intervals in the complement. In case that intervals fall into intervals, the complete number can be reduced. In no case it can grow. Therefore the assertion of uncountably many degenerate intervals (so called singletons - but there cannot exist irrational singletons without rational numbers separating them) in the complement has been contradicted.

§ 093 The difficulty we are confronted with is that ZFC makes a claim we find implausible. To say we can't criticize ZFC since ZFC is our theory of sets is obviously to beg the question whether we ought to adopt it despite claims about cardinality that we might regard as exorbitant. [George Boolos: "Must We Believe in Set Theory?"]

Cancer robbed our community of an outstanding philosopher. One is tempted to say “philosopher and logician,” but as Richard Cartwright remarked in his eulogy for George Boolos, “he would have not been altogether happy with the description: accurate, no doubt, but faintly redundant - a little like describing someone as ‘mathematician and algebraist.’”

[...] George Boolos made significant contributions in every area of logic in which he worked. [...] Gödel, in contrast to Boolos, argued that the axioms of choice and replacement do follow from the iterative conception. In article 7, an introduction to a posthumously published lecture by Gödel, Boolos takes issue with Gödel’s Platonistic claim that the axioms of ZFC (Zermelo Frankel set theory with Choice) “force themselves upon us as true.” Even if the axioms articulate a natural and compelling conception of set, they need not correspond to anything objectively real.

Article 2 contains Boolos’ defense of Fraenkel’s, in contrast to Zermelo’s, position that first-order but not second-order logic is applicable to set theory. Boolos criticizes the view of Charles Parsons (and D. A. Martin) that it makes sense to use second-order quantifiers when first-order quantifiers range over entities that do not form a set. Boolos’ answer to the title of article 8, “Must We Believe in Set Theory?” is "no": the phenomenological argument (due to Gödel) does not imply that the axioms of set theory correspond to something real, and the indispensability argument (due to Carnap) that mathematics is required by our best physical theory, is dismissed as “rubbish.”

[Gary Mar: "Book Review: Logic, Logic and Logic, George Boolos. Harvard University Press, 1998. ix + 443 pages. Hardcover $45, paperback $22.95. ISBN 0-674-53767-X", Essays in Philosophy, Vol. 1 No. 2, June 2000] http://commons.pacificu.edu/cgi/viewcontent.cgi?article=1013&context=eip

§ 094 Nicole d'Oresme (1323 - 1382) proved the harmonic series to be divergent. Alas he needs ¡0 sums of the form

(1/2) + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + (1/9 + ... + 1/16) + ... with in total 2¡0 unit fractions. If there are less than 2¡0 natural numbers, then there are also less than 2¡0 unit fractions and the proof of Nicole d'Oresme will fail.

But 2¡0 elements are countable, as is easily proved by the nodes of the Binary Tree:

0 1, 2 3, 4, 5, 6 7, ...

¡0 The bijection proves 2 = ¡0. There is but one problem: The "limit-level" of the Binary Tree does not contain any numbered node. The natural numbers are all exhausted before. (For a definition of limit by liminf and limsup cp. § 090.) That means, the paths of the Binary tree, i.e. the real numbers of the unit interval, cannot differ "in the limit". Can they be distinguished at all?

§ 095 Dr. D.F.M. Strauss, is professor of philosophy at the University of the Free State in Bloemfontein, South Africa.

In Chapter II Strauss addresses various philosophical problems in mathematics. Mathematics is concerned with number and space, the first two modalities. The prime issue in mathematics is how to treat infinity. Strauss discusses three main foundational crises in the history of mathematics: (1) the discovery of irrationals, (2) infinitesimal calculus, and (3) modern set theory. All three involve the relation between potential and actual infinity. Much attention is devoted to the conflict between Cantor's treatment of actually infinite sets and the intuitionists' rejection of actual infinity. The Dutch mathematician L.E.J. Brouwer (1882-1966), an ardent promoter of intuitionism, lived in Amsterdam at the same time as Dooyeweerd and had some influence on Dooyeweerd. Dooyeweerd acknowledged only the potential infinite; he found the idea of the actual infinite unacceptable. {{Obviously there are many people sharing this opinion. Unfortunately modern censorship cares that these people rarely get a chance to write in mathematical journals. Compare the situation with Hilbert, who fired Brouwer from the board of the , starting what Einstein called the war of the frogs and the mice. For this Machtergreifung in 1928 Hilbert had less authorization than Hitler had for his in 1933.}} Strauss, however, argues that Dooyeweerdian philosophy actually provides grounds for both types of infinity. Strauss distinguishes between the successive infinite and the at once infinite. The successive infinite is associated with numbers and determines every denumerable, endless succession of numbers (e.g., the integers or rational numbers). The at once infinite, on the other hand, is associated with the continuous extension of space. The latter represents a higher order of infinity; it cannot be reduced to a successive infinity since space cannot be reduced to number {{that's why uncountable sets are an absurdity.}}

Review: D.F.M. Strauss "Paradigms in Mathematics, Physics and Biology: Their Philosophical Roots", Tekskor Bk, Danhof, South Africa, (2001, revised 2004), 177 pp.

§ 096 It is true, I believe, that the meaning of an expression is a matter of how it is used. But this does not mean that meaning can be reduced to use. Rather, meaning permeates use. We manifest our understanding of expressions in our use of them. We communicate with others by shared access to one another's use of them. There is nothing here to suggest that we should always be able to "pin down" what an expression means. [...] How then do people manage to grasp such meaning? Well, how do they? They observe expressions being used. They try to see the point of the use. They try to use the expressions in the same way, under the guidance of promptings, corrections, and encouragement from others. Yes, but if there is nothing in how an expression is used which they can have access to before understanding it and which actually serves to determine the expression's (full infinite) meaning, how does any of this help? Will they not be confronted by something which strikes them as being, at best, radically inconclusive and, at worst, so much incomprehensible babble? Initially perhaps. But they eventually come to understand. It is true that this can seem quite mysterious. What we have to do, however, is to see it as perfectly natural. People just do have shared interests, and a shared sense of what is significant and of how things relate to one another. (These are partly innate and partly inculcated.) As a result, people are able to understand one another. They are able to see what other people are up to. They are able to grasp what expressions mean. In the mathematical case, there is no reason why being subjected to (some of) the truths of a formal theory - seeing how these truths are proved and the kinds of justifications that are proffered for them - should not give someone a sense of how to carry on, even though not all the truths have been, or could be, captured. I am leaning heavily at the moment on some of Wittgenstein's later work on meaning and language use. [...] Wittgenstein was regarded by many as one of the chief architects, along with Russell, of what became known as logical . [...] "What we cannot speak about," he wrote in conclusion, "we must pass over in silence." However - and this is the twist - what we cannot speak about, or what cannot be said, can, Wittgenstein maintained, be shown: the nonsense in the Tractatus had arisen from an attempt to put genuine insights into words. This distinction between what can be said and what can be shown - the saying/showing distinction - was a linchpin of the whole book. No feature of the world as a whole could properly be conveyed in words. The framework in which all the facts were held together was not itself a fact. Features of the world as a whole, its overall shape and form, were a matter not of its being how it was but of its being however it was. They were a matter not of what could be said but of what was involved in saying anything at all. They were what could be shown. [A.W. Moore: “The Infinite”, 2nd ed., , New York (2005), p. 183-188]

This should be heeded by addicts of the formal-definitions obsession, in particular Bourbaki- Dieudonné and others of that ilk.

§ 097 According to the status quo, the continuum is properly modelled by the "real numbers". What is a real number? Let's start with an easier question: What is a rational number? Here comes set theory to our aid. It is, according to some accounts, nothing but an equivalence class of ordered pairs of integers. Thus when my six year old daughter uses the fraction what she is really doing is using the "equivalence class" [...] Sequences generated by algorithms can be specified by those algorithms, but what possibly could it mean to discuss a "sequence" which is not generated by such a finite rule? Such an object would contain an "infinite amount" of information, and there are no concrete examples of such things in the known universe. This is metaphysics masquerading as mathematics. [N J Wildberger: "Set Theory: Should You Believe?"] http://web.maths.unsw.edu.au/~norman/views2.htm

§ 098 The critical analysis which has given physics its new confidence [Einstein's "operational" technique] has, up to the present, been almost exclusively confined to an examination of the nature of the physical concepts which the physicist uses. But since mathematics is coming to play an increasingly important role in the new physics, it is evident that a critical examination of the nature of the fundamental concepts of mathematics is a task of the immediate future for the physicist. It was therefore not without a certain amount of dismay that I suddenly became aware that in the mathematics of the present day there are doubts, uncertainties, and differences of opinion on fundamental questions which are at least not unlike the bewilderment of physics when confronted with the new phenomena of the quantum domain. My awakening to a consciousness of the situation in mathematics I owe to that extraordinary well written little book by E.T. Bell, "The Queen of the Sciences". Within a few weeks of my reading this book A. F. Bentley's "Linguistic Analysis of Mathematics appeared. This I skimmed hastily, gathering from it a most vivid impression of the chaotic state of affairs in the "fundamental" fields of mathematics [...] These two expositions made it evident that Mengenlehre was that branch of mathematics in which perhaps there were the most serious differences of opinion and in which fundamental questions were most to the fore. [...] The reactions to this second acquaintance with Mengenlehre were as different as can well be imagined from those of my first naive contact, when I suppose I got the usual kick out of feeling that I was playing with the infinite. [...] the meaning of a term or concept is contained in those operations which are performed in making application of the term or concept to relevant situations. I can only report that as a matter of personal analysis I find this operational aspect at the bottom of all meaning, but my is that other persons go through similar processes [...] we may perhaps say that self-consistency is in some way intimately connected with real things. [...] The accepted method of proving that some system of postulates does not conceal some contradiction is to exhibit some "real", "existing" system which satisfies the postulates. Nothing further in the way of proof or analysis is felt to be necessary; the feeling that actually existing things are not self-contradictory is so elemental as almost to constitute a definition of what we mean by self-consistent. Now when we are concerned with "things" we are evidently concerned with some form of experience, so that we may make an even broader statement and say that experience is not self-contradictory. [...] it is once obvious that the operational technique automatically secures to mathematics the sine qua non of self-consistency, for operations actually carried out, whether physical or mental, are a special form of experience, so that any mathematical concept or argument analyzed into actual operations must have the self-consistency of all experience. [...] "point" has no meaning unless it is defined, and this involves the specification of some sort of procedure. "All the points of a line" as a purely intuitional concept apart from the rules by which points are determined, can have no operational meaning, and accordingly is to be held for mathematics as an entirely meaningless concept. [...] "All the points of a line" means no more than "All the rules for determining points on a line" {{which are known to form a countable set}}. [...] In other words, we have no more reason to describe the points on a line as non- denumerable than the non-terminating decimals. The repudiation of the diagonal Verfahren for the decimals at the same time removes all reason for thinking the points on a line non- denumerable. In fact, a consistent application of the operational criterion of meaning appears to demand the complete discard of the notion of infinities of different orders. We never have "actual" infinites [...] but only rules for operation which are not self-terminating. How can there be different sorts of non-self-terminatingness? At any stage in the process the rule either permits us to go on and take the next step or it does not [...] and that is all there is to it. [...] Mengenlehre is similarly supposed to have established the existence of by showing that all algebraic numbers are denumerable. This proof I would reject, holding that the mere act of assigning operational meaning to the transcendentals of itself ensures that they are denumerable. As a matter of fact, only a few transcendentals have been established. Mengenlehre is powerless to show whether any given number, such as e or π, is transcendental or not, and the detailed analysis necessary in any given case for establishing transcendence is not avoided by Mengenlehre. From the operational view a transcendental is determined by a program of procedure of some sort; Mengenlehre has nothing to add to the situation. And this, as far as my elementary reading goes, exhausts the contributions which Mengenlehre has made in other fields. [P.W. Bridgman: "A physicist's second reaction to Mengenlehre", Scripta Mathematica, Vol. II, 1934]

We should repeat this remarkable sentence: "Mengenlehre has nothing to add to the situation. And this, as far as my elementary reading goes, exhausts the contributions which Mengenlehre has made in other fields." But we can emphasize this remarkable sentence, also from a less elementary point of view. Walter Felscher http://de.wikipedia.org/wiki/Felscher autor of a text book on set theory in three volumes writes: "Concerning the application of transfinite numbers in other mathematical disciplines {{outside of mathematics nobody claims any use of set theory (for the always present exceptions of the rule compare Kalenderblatt 120422 to 120429)}}, the great expectations originally put on set theory have been fulfilled only in few special cases ..." [Walter Felscher: "Naive Mengen und abstrakte Zahlen III", Bibl. Inst., Mannheim (1979) p. 25] http://books.google.de/books/about/Naive_Mengen_und_abstrakte_Zahlen.html?id=s0c6AAAA MAAJ&redir_esc=y

§ 099 Alexander Zenkin: OPEN LETTER TO: The Bulletin of Symbolic Logic CC: The International Mathematical Union

Dear Professor Blass,

As you certainly guessed, the question is not only about a publication of my comments "Whether the Lord Exists in G. Cantor's Transfinite 'Paradise'" to the scandally-known, quasi- "pedagogical" W. Hodges' paper "An Editor Recalls Some Hopeless Papers" [...] in your BSL- journal. The question is about much more important problems. I believe that BSL-papers like the W. Hodges' one are a dangerous phenomenon from scientific, educational, and social points of view. There are the following reasons to state that. 1. The main conclusion of the W. Hodges' paper that "there is nothing wrong with Cantor's argument" is wrong fundamentally and therefore, having been proclaimed in the BSL, such a conclusion disorients a wide scientific community (pedagogical, mathematical, logical, philosophical, cognitive psychological, etc. ones) as to one of the most important problems of the humankind culture as a whole - the problem on the veritable nature of Infinity. 2. The high symbolic logic level of the BSL-publications is a recognized standard of a meta- mathematical thinking and an attractive for young generations of meta-mathematicians and symbolic logicians. However a one-sided publishing BSL-policy (not to publish points of view differing from the traditional set theoretical opinion) deprives the young generation of the democratic right to make independently its own scientific choice between two historical, contradictory points of view as to the true nature of Infinity: i.e., between the today traditional Cantor's and all modern axiomatic set theory's opinion, on the one hand, and the opinion of Aristotle, Leibniz, Kant, Gauss, Cauchy, Kronecker, Hermite, Poincaré, Bair, Borel, Brouwer, Wittgenstein, Weil, Luzin, Quine, and today - Sol. Feferman, Ja. Peregrin, V. Turchin, P. Vopenka, etc. [...] 3. One of the most important reasons for my flat objection against the W. Hodges' and similar meta-mathematical papers is their deforming influence on mathematical education and their dangerous social consequences as a whole. As far back as the middle of 50s of the XX century, the outstanding American mathematician, [...] warned: "Too much formalization and symbolization in the theory of mathematics is dangerous for the healthy development of the science of mathematics". In the beginning of the 60s, a large group (about 75) of outstanding mathematicians of America and Canada (including Richard Bellman of Rand Corporation, Richard Courant of New York University, Н.О. Pollak of Bell Telephone Laboratories, George Polya of Stanford University, Andre Weil of Institute for Advanced Study, and others) tried to attract attention of mathematical community to the same problem - to the danger to provoke a stable disgust of pupils, students and their parents (who, by the way, are today's Presidents, Government-men, Congressmen, Government ministers, etc.) to mathematics by means of a premature, excessive, deterrent, and simply thoughtless formalization of mathematical education. In their known Memorandum "ON ТНЕ MATHEMATICS CURRICULUM OF THE HIGH SCHOOL" (American Mathematical Monthly, 1962, March, 189-193) they, in particular, wrote. "It would [...] bе а tragedy if the curriculum reform [...] should be misdirected and the golden opportunity wasted. There are, unfortunately, factors and forces in the current scene which may lead us astray. [...] premature formalization may lead to sterility; premature introduction of abstractions meets resistance especially from critical minds who, before accepting an abstraction, wish to know why it is relevant and how it could be used. In its cultural significance as well as in its practical use, mathematics is linked to the other sciences and the other sciences are linked to mathematics, which is their language and their essential instrument. Mathematics separated from the other sciences loses one of its most important sources of interest and motivation. [...] We wish especially that the new curricula should reflect more the connection between mathematics and science and carefully heed the distinction between matters logically prior and matters which should have priority in teaching. Only in this way can we hope that the basic values of mathematics, its real meaning, purpose, and usefulness will be made accessible to all students [...]" In conclusion, they again accentuate and expressed their "concern about а trend to excessive emphasis on abstraction in the teaching of mathematics" As the posterior history showed, this very serious, very anxious, and high professional warning of outstanding mathematicians of the middle of the XX c. as to the danger of the "excessive formalization and symbolization of mathematical education" was not heard. Today the situation is further aggravated. The Vice-President of the International Mathematical Union, Academician of the Russian Academy of Sciences, outstanding mathematician and mathematical educator, professor Vladimir I. Arnold of Steklov Mathematical Institute (Moscow) in his numerous papers of the last decade again tries hard to attract attention of mathematicians and educational community to the catastrophic situation in modern mathematics and mathematical education. The main reason is the same one – a (today already) global super- formalization or, using his term, “bourbakization” [...] of the modern mathematics as a whole (see, e.g., V.I. Arnold, "International Mathematical Congress in Berlin." [...]) “Our brain, - writes Arnold, - has two halves: one is responsible for the multiplication of polynomials and languages , and the other half is responsible for orientation of figures in space and all the things important in real life . Mathematics is geometry when you have to use both halves. In the middle of the XX Century, a [...] mafia of “left-hemispheric” mathematicians could exclude geometry from mathematical education (firstly in France, and then in other countries), replaced all informal aspects of this discipline by a training in a formal manipulation with abstract “notions” . All geometry, and consequently all connection of mathematics with the real world and with other sciences was excluded from mathematical education. Such the “abstract” description of mathematics is unfit neither for education, nor for any practical applications. …Compelling miserable schoolboys/girls to learn such , “left- hemispherical criminals” created a modern distinctly negative attitude of society and governments to mathematics. …The aversion to mathematics which government ministers, exposed to such the experience of such the education in school, have is a healthy and valid reaction. Unfortunately, this their disgust spreads on all mathematics without exclusions, and that can kill it as a whole”. ... these “left-hemispherical invalids” were able to cultivate whole generations of mathematicians that don’t understand any other approach to mathematics and are able only to teach next generations by the same way. … It is awful to think what kind of pressure the Bourbakists put on (evidently nonsilly) students to reduce them to formal machines! This kind of formalized education is completely useless for any practical problem and even dangerous, leading to Chernobyl-type accidents. [...] … Modern formalized (bourbakized) education in mathematics is an exact antithesis for teaching the and the true scientific foundations of mathematics. Such the mathematical education is dangerous for a humankind as a whole”. The “clinical” picture of the “bourbakism” drawn by Prof. Arnold verily can be called a mental Acquired Immunodeficiency Syndrome of brain, i.e., shortly a MENTAL-AIDS. As an experience testifies, the MENTAL-AIDS is a very infectious illness which affects especially easy an unprotected kid’s brain, unfortunately, without any perspective to get well: as far back as XVIII c. the great English philosopher G. Berkeley said that "a human-being mind, immersed in high level abstractions from a young age, loses an adequate perception of the real world to its adult age". I shouldn't be surprised if many parents of modern schoolboys/girls and students would like to bring an action against modern Cantorians and their official "scientific" communities and journals because of their deliberate cultivation and propagation of such the dreadful infectious social disease as that MENTAL-AIDS. However that may be, today, in the very beginning of the XXI Century, we have a much more painful diagnosis concerning prospects of modern mathematics and mathematical education. I state and can prove that the main historical source of this dangerous social illness is just the modern cantorianism with its pure abstract, ambitious transfinite constructions with an empty , based upon the only Cantor's theorem on the uncountability of real numbers. [...] I am sure that there is a lot of judicious mathematicians and simply provident parents of future mathematicians of genius who would not like that their children became "formal machines" used to execute criminal, terroristic, anti-human "deductive" plans. I hope to have their active support. Nobody, including the BSL-team, will save the Cantor's transfinite "paradise".

Sincerely yours, Alexander Zenkin [...]

P.S.1. I have attentively read the enclosed BSL-reviewer's report [...] and regret too that the report reviewer distorted the sense of the Comments-1 deliberately, high professionally and fundamentally, and misled you and the mathematical and symbolic logic community as to the important problems touched upon in the Comments-1. [...]

P.S.2. [...] my system VISAD (for VISual Anaysis of Data), based on the Cognitive Computer Graphics (CCG) conception, has fulfilled a comparative analysis of CCG-images of the W. Hodges' paper text and the anonymous BSL-reviewer's report text and has established that the both authors are the same face . - It is quite interesting result from the professional scientific point of view, is not so? http://www.ccas.ru/alexzen/papers/OPEN_LETTER-2_to_the_BSL.doc

§ 100 The pure mathematician can do what he pleases, but the applied mathematician must be at least partially sane. [J. Willard Gibbs, quoted in Morris Kline: "Mathematics: The Loss of Certainty", Oxford University Press (1980) p. 285] http://books.google.de/books?id=RNwnUL33epsC&pg=PA285&lpg=PA285&dq=%22The+pure+ mathematician+can+do+what+he+pleases%22&source=bl&ots=P84nY_76Mg&sig=6qOnuCW2 RzyzU8vBgoqPIpR8AkA&hl=de&sa=X&ei=zCIQULroN8X- 4QTFkIGwCQ&sqi=2&ved=0CFMQ6AEwAw#v=onepage&q=%22The%20pure%20mathematici an%20can%20do%20what%20he%20pleases%22&f=false

§ 101 The first virtue of a matheologian is the ability to anaesthetize himself (or herself) against the contradictions arising from matheology.

§ 102 Our contemporary orthodoxy: to show that there are so-and-sos is to prove "so-and-sos exist" from the axioms of set theory. [Penelope Maddy: "Mathematical Existence", Bull. Symbolic Logic 11 (2005) 351] http://www.jstor.org/pss/1578738 Note: contemporary means not timeless.

§ 103 Today I received by mail an offprint of "Statements concerning the theory of the transfinite" with a handwritten dedication: "H. A. Schwarz in memory of our old friendship dedicated by the author." After having had the opportunity to go through it leisurely, I cannot conceal that it appears to me as a pathological aberration. What on earth have the Fathers of the Church to do with the irrational numbers? I really hope my fear might not come true, that our patient has left the straight and narrow like the poor Zöllner* who never found the way back to scientific business. The more I think over these cases the more I am forced to get aware of the similarity of symptoms. Might we manage to lead the poor young man back to serious work! Otherwise it will come to a bad end with him. [H.A. Schwarz to C. Weierstraß, Oct. 17, 1887] *) Johann Karl Friedrich Zöllner (1834-1882) was a professor of astrophysics who later got involved in depth in philosophical studies and after all became an adherent of spiritism. German original text from Herbert Meschkowski: "Georg Cantor: Leben, Werk und Wirkung", 2nd ed., Bibl. Inst., Mannheim (1981) pp. 266-267 can be found in: Das Kalenderblatt 091019 http://www.hs-augsburg.de/~mueckenh/KB/KB%20001-200.pdf

§ 104 I cannot consider the set of positive integers as given, for the concept of the actual infinite strikes me as insufficiently natural to consider it by itself. (Nikolai Nikolaevich Luzin to Kazimierz Kuratowski) [Naum Yakovlevich Vilenkin: "In search of infinity", Birkhäuser, Boston (1995) p. 126] http://books.google.de/books?id=cU3HQFek7L0C&printsec=frontcover&source=gbs_v2_summa ry_r&cad=0#v=onepage&q=&f=false

{{Obvious for matheologians:}} For Luzin this was torture. His progress in mathematics at the Gymnasium became worse and worse, so that his father was obliged to engage a tutor ... [J.J. O'Connor, E.F. Robertson: Luzin Biography, Mac Tutor] http://www-history.mcs.st-and.ac.uk/Biographies/Luzin.html

§ 105 What if I were to travel back in time and kill my past self? If my past self died, then there would be no I to travel back in time, so I wouldn't kill my past self after all. So then the time-trip would take place, and I would kill my past self. And so on. I was also disturbed by the fact that if the future is already there, then there is some sense in which our is an illusion. Gödel seemed to believe that not only is the future already there, but worse, that it is, in principle, possible to predict completely the action of some given person. {{There he believed in completeness?}} I objected that if there were a completely accurate theory predicting my actions, then I could prove the theory false-by learning the theory and then doing the opposite of what it predicted. According to my notes, Gödel's response went as follows: "It should be possible to form a complete theory of human behavior, i.e., to predict from the hereditary and environmental givens what a person will do. However, if a mischievous person learns of this theory, he can act in a way so as to negate it. Hence I conclude that such a theory exists, but that no mischievous person will learn of it. In the same way, time-travel is possible, but no person will ever manage to kill his past self." Gödel laughed his laugh then, and concluded, "The a priori is greatly neglected. Logic is very powerful." Apropos of the free will question, on another occasion he said: "There is no contradiction between free will and knowing in advance precisely what one will do. If one knows oneself completely then this is the situation. One does not deliberately do the opposite of what one wants." {{There exists free will, but nobody uses it to decide alternatively. - No contradiction.}} [...] Gödel's credo, "I do objective mathematics." By this, Gödel meant that mathematical entities exist independently of the activities of mathematicians, in much the same way that the stars would be there even if there were no astronomers to look at them. For Gödel, mathematics, even the mathematics of the infinite, was an essentially empirical science. [...] Cantor's Continuum Problem is undecidable on the basis our present-day theories of mathematics. For the formalists this means that the continuum question has no definite answer. But for Platonist like Gödel, this means only that we have not yet "looked" at the continuum hard enough to see what the answer is. [...] the same possibilities of thought are open to everyone, so that we can take the world of possible forms as objective and absolute. Possibility is observer- independent, and therefore real, because it is not subject to our will. [...] anyone who takes the trouble to learn some mathematics can "see" the set of natural numbers for himself. So, Gödel reasoned, it must be that the set of natural numbers has an independent existence, an existence as a certain abstract possibility of thought. I asked him how best to perceive pure abstract possibility. He said three things. i) First one must close off the other senses, for instance, by lying down in a quiet place. It is not enough, however, to perform this negative action, one must actively seek with the mind. ii) It is a mistake to let everyday reality condition possibility, and only to imagine the combinings and permutations of physical objects - the mind is capable of directly perceiving infinite sets. iii) The ultimate goal of such thought, and of all philosophy, is the perception of the Absolute. Gödel rounded off these comments with a remark on Plato: "When Plautus could fully perceive the Good, his philosophy ended." Gödel shared with Einstein a certain mystical turn of thought. {{Gödel may have been a great mythologican. Einstein's known legacy does not support Rucker's claims - on the contrary: http://www.hillmanweb.com/reason/inspiration/einstein.html }} The central teaching of mysticism is this: Reality is One. The practice of mysticism consists in finding ways to experience this higher unity directly. The One has variously been called the Good, God, the , the Mind, the Void, or (perhaps most neutrally) the Absolute. No door in the labyrinthine castle of science opens directly onto the Absolute. But if one understands the maze well enough, it is possible to jump out of the system and experience the Absolute for oneself. {{Selfexperience of the Absolute is the essential element of every religion.}} [...] I asked Gödel if he believed there is a Single Mind behind all the various appearances and activities of the world. He replied that, yes, the Mind is the thing that is structured, but that the Mind exists independently of its individual properties. I then asked if he believed that the Mind is everywhere, as opposed to being localized in the brains of people. Gödel replied, "Of course. This is the basic mystic teaching." We talked a little set theory, and then I asked him my last question: "What causes the illusion of the passage of time?" Gödel spoke not directly to this question. [...] He went on to relate the getting rid of belief in the passage of time to the struggle to experience the One Mind of mysticism. Finally he said this: "The illusion of the passage of time arises because we think of occupying different . In fact, we occupy only different givens. There is only one reality." I wanted to visit Gödel again, but he told me that he was too ill. In the middle of January 1978, I dreamed I was at his bedside. There was a chess board on the covers in front of him. Gödel reached his hand out and knocked the board over, tipping the men onto the floor. The chessboard expanded to an infinite mathematical plane. And then that, too, vanished. There was a brief play of symbols, and then emptiness - an emptiness flooded with even white light {{confirming the intercerebral existence form of the mind and thereby the possible existence of a mindscape capable of absorbing uncountable numbers of numbers and ideas. (The rocks of the Moon were there before the lunar module landed; and all the possible thoughts are already out there in the Mindscape. An idea is already there in the Mindscape whether or not someone is thinking it. (Rucker))}} The next day I learned that Kurt Gödel was dead {{In order to make the most of this mysterious text: finally imitate (or imagine - according to your skills) the call of the screech owl.}} [Rudy Rucker: "Infinity and the Mind", Princeton University Press, Princeton (2005) pp. 36, 168ff]

§ 106 Views to the effect that Platonism is correct but only for certain relatively "concrete" mathematical "objects". Other mathematical "objects" are man made, and are not part of an external reality. Under such a view, what is to be made of the part of mathematics that lies outside the scope of Platonism? An obvious response is to reject it as utterly meaningless. [Harvey M. Friedman: "Philosophical problems in logic" (2002) p. 9] http://www.math.osu.edu/~friedman.8/pdf/Princeton532.pdf

§ 107 Cantor had a different set of numinous feelings about the infinite. He was not only a great mathematician, but a very religious man and by some standards a mystic. Yet his mysticism was supported by his mathematics, which to him was at least as strong an argument for the mathematics as for the mysticism. Apart from claiming divine inspiration for his work, we don't know exactly what spiritual views he linked to his mathematics, but his theorems give support to the following. Measured in meters, we are tiny specks compared to the universe at large. But measured in dimensionless points, we are as large as the universe: a proper subset, but one with the same cardinality as the whole. Similarly, measured in meters, we may be off in a corner of the universe. But measured in points, the distance is equally great in all directions, whether universe is finite or infinite; that puts us in the center, wherever we are. Measured in days, our lives are insignificant hiccups in the expanse of past and future time. But measured in points of time, our lives are as long as universe is old. We are as small as we seem, but simultaneously, by a most reasonable measure, co-extensive with the totality of being in both space and time. {{These few properties already should allow the average rational scientist (as opposed to the mystic matheologian) to recognize how "reasonable" this measure is.}} [Peter Suber: "Infinite Reflections", St. John's Review, XLIV, 2 (1998)] http://www.earlham.edu/~peters/writing/infinity.htm#sublimity

§ 108 The main part of the paper is devoted to show that the real numbers are denumerable. The explicit denumerable sequence that contains all real numbers will be given. The general element that generates the sequence will be written as well as the first few elements of that sequence. That there is one-to-one correspondence between the real numbers and the elements of the explicitly written sequence will be proven by the three independent proofs. [...] It is also proven that the Cantor’s 1873 proof of non denumerability is not correct since it implicates non denumerability of rational numbers. In addition it is proven that the numbers generated by the diagonal procedure in Cantor’s 1891 proof are not different from the numbers in the assumed denumerable set. [S. Vlahovica, B. Vlahovic: "Countability of the Real Numbers", arXiv:math/0403169 (2004)] http://arxiv.org/abs/math/0403169

§ 109 Cantor’s diagonal method has been the proof for the uncountable infinite set of real numbers to be “larger” than the countable infinite set of natural numbers. In the following work Cantor’s method is refuted and it is proven that the cardinality of real numbers is the same as the cardinality of natural numbers. Contrary to Cantor’s method, this proof is constructive and estimates directly the cardinality of the real numbers and compares it to the natural numbers by constructing an injection to the prime numbers. The work is written in German and shall be translated into English. Until then the German version is the only source for the proof. [J. Grami, A. Grami: "Die reellen Zahlen sind abzählbar" (2009)] www.real-numbers.de

§ 110 Belief in the existence of the infinite comes mainly from five considerations:

(1) From the nature of time - for it is infinite. (2) From the division of magnitudes - for the mathematicians also use the notion of the infinite.

Further, how can the infinite be itself any thing, unless both number and magnitude, of which it is an essential attribute, exist in that way? If they are not substances, a fortiori the infinite is not. It is plain, too, that the infinite cannot be an actual thing and a substance and principle.

This discussion, however, involves the more general question whether the infinite can be present in mathematical objects and things which are intelligible and do not have extension, as well as among sensible objects. Our inquiry (as physicists) is limited to its special subject-matter, the objects of sense, and we have to ask whether there is or is not among them a body which is infinite in the direction of increase. We may begin with a dialectical argument and show as follows that there is no such thing. If 'bounded by a surface' is the definition of body there cannot be an infinite body either intelligible or sensible. Nor can number taken in abstraction be infinite, for number or that which has number is numerable. If then the numerable can be numbered, it would also be possible to go through the infinite.

It is plain from these arguments that there is no body which is actually infinite. But on the other hand to suppose that the infinite does not exist in any way leads obviously to many impossible consequences: there will be a beginning and an end of time, a magnitude will not be divisible into magnitudes, number will not be infinite. If, then, in view of the above considerations, neither alternative seems possible, an arbiter must be called in; and clearly there is a sense in which the infinite exists and another in which it does not. We must keep in mind that the word 'is' means either what potentially is or what fully is. Further, a thing is infinite either by addition or by division. Now, as we have seen, magnitude is not actually infinite. But by division it is infinite. (There is no difficulty in refuting the theory of indivisible lines.) The alternative then remains that the infinite has a potential existence.

The infinite exhibits itself in different ways-in time, in the generations of man, and in the division of magnitudes. For generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite, but always different.

But in the direction of largeness it is always possible to think of a larger number: for the number of times a magnitude can be bisected is infinite. Hence this infinite is potential, never actual: the number of parts that can be taken always surpasses any assigned number. But this number is not separable from the process of bisection, and its infinity is not a permanent actuality but consists in a process of coming to be, like time and the number of time. With magnitudes the contrary holds. What is continuous is divided ad infinitum, but there is no infinite in the direction of increase. For the size which it can potentially be, it can also actually be. Hence since no sensible magnitude is infinite, it is impossible to exceed every assigned magnitude; for if it were possible there would be something bigger than the heavens.

Our account does not rob the mathematicians of their science, by disproving the actual existence of the infinite in the direction of increase, in the sense of the untraversable. In point of fact they do not need the infinite and do not use it. They postulate only that the finite straight line may be produced as far as they wish. It is possible to have divided in the same ratio as the largest quantity another magnitude of any size you like. Hence, for the purposes of proof, it will make no difference to them to have such an infinite instead, while its existence will be in the sphere of real magnitudes.

It remains to dispose of the arguments which are supposed to support the view that the infinite exists not only potentially but as a separate thing. Some have no cogency; others can be met by fresh objections that are valid.

[Aristoteles: "Physics, Book III", Part 4 (350 v. Chr.)] http://www.greektexts.com/library/Aristotle/Physics/eng/index.html http://classics.mit.edu/Aristotle/physics.3.iii.html

§ 111 Let m and n be two different characters, and consider a set M of elements

E = (x1, x2, ..., xν, ...) which depend on infinitely many coordinates x1, x2, ..., xν, ..., and where each of the coordinates is either m or w. Let M be the totality of all elements E. To the elements of M belong e.g. the following three:

EI = (m, m, m, m, ...), EII = (w, w, w, w, ...), EIII = (m, w, m, w, ...).

I maintain now that such a manifold M does not have the power of the series 1, 2, 3, ..., ν, ... . This follows from the following proposition: "If E1, E2, ..., Eν, ... is any simply infinite series of elements of the manifold M, then there always exists an element E0 of M, which is not equal to any element Eν." For proof, let there be

E1 = (a1,1, a1,2, ..., a1,ν, ...) E2 = (a2,1, a2,2, ..., a2,ν, ...) ... Eμ = (aμ,1, aμ,2, ..., aμ,ν, ...) ... where the characters aμ,ν are either m or w. Then there is a series b1, b2, ..., bν, …, defined so that bν is also equal to m or w but is different from aν,ν. Thus, if aν,ν. = m, then bν = w, and if aν,ν. = w, then bν = m. Then consider the element

E0 = (b1, b2, b3, ...) of M, then one sees straight away, that the equation

E0 = Eμ cannot be satisfied by any positive integer μ, otherwise for that μ and for all values of ν

bν = aμ,ν and so we would in particular have

bμ = aμ,μ which through the definition of bν is impossible. From this proposition it follows immediately that the totality of all elements of M cannot be put into the sequence: E1, E2, ..., Eν, ... otherwise we would have the contradiction, that a thing E0 would be both an element of M, but also not an element of M.

[G. Cantor: "Über eine elementare Frage der Mannigfaltigkeitslehre", Jahresbericht DMV I (1890-91) 75-78]

A proof by contradiction fails, if only one counter example can be found. Here it is:

Consider the sequence

E1 = (w, m, m, m, m, m, m, ...) E2 = (m, w, m, m, m, m, m, ...) E3 = (m, m, w, m, m, m, m, ...) E4 = (m, m, m, w, m, m, m, ...) ...

This matrix formed by the aμ,ν has no line μ and no column ν with all characters aμ,ν = m. Since such a line or column would need infinitely many predecessors, namely all lines or columns with a finite number of m before the w, it cannot belong to the sequence. (It is the limit of the sequence.)

Define E0 = (b1, b2, b3, ...) by bμ = m ∫ w = aμ,μ.

The first μ characters of E0 agree with the first μ characters of all Eν for all ν > μ. Since there is no last μ and no last ν, this situation does never change. Otherwise we would have the contradiction that a matrix has more* characters m on the diagonal than it has in any line and in any column**. *) i.e. a number of m before the first w that is larger than every finite number. **) where the number of m before the first w is always finite.

§ 112 There is no evidence that Cantor himself ever considered the possibility that the continuum hypothesis is undecidable. Obviously formal investigations were far from his mind. [p. 213] For him mathematical theorems were theses about something being; he even was convinced that the cardinal numbers ¡0 and ¡ were corresponding to realities in the physical world. We are afraid, he would not have enjoyed the "solution" of his questions by the modern foundational researcher. [p. 213] {{This sentence could be improved by another pair of quotation marks.}} Obviously it was difficult for Cantor to express in hard mathematical language, what he imagined. His "definitions" could appear rather questionable to a critical thinker like Kronecker. [p. 229] {{Good intuition is certainly preferable over formalism.}} I have no doubts concerning the truth of the transfinite that I have recognized by help of God and have been studying in its diversity and unity for more than 20 years. Every year and nearly every day advances me in this science. [...] From no other subjects of created nature I have safer knowledge than of the theorems of transfinite number- and type-theory. [G. Cantor to Pater I. Jeiler, Whitsun 1888] {{Of course this intuition must not fail.}} Or is it advisable to completely refrain from set theory in primary school? Today we tend to recommend that. [p. 227] {{And not only there!}} [Quotations with page numbers from Herbert Meschkowski: "Georg Cantor: Leben, Werk und Wirkung", 2nd ed., Bibl. Inst., Mannheim (1981)] For German original texts see: Das Kalenderblatt 100707 http://www.hs-augsburg.de/~mueckenh/KB/KB%20201-400.pdf

§ 113 [...] gradually and unwittingly mathematicians began to introduce concepts that had little or no direct physical meaning. [...] concepts and theories that do not have immediate physical interpretation [...] gained acceptance. [...] The gradual rise and acceptance of the view that mathematics should embrace arbitrary structures that need have no bearing, immediate or ultimate, on the study of nature led to a schism that is described today as pure versus applied mathematics. [1] "The Fall and Original sin of set theory" [2], leads to matheology that praises a mixture of "mush of the continuum" (Brei des Kontinuums) with "space sauce" (Raumsauce) [3] as exquisite brainfood, but nevertheless is not ashamed to call the real numbers a set. Sets consist of distinguishable elements. [1] M. Kline: "Mathematical Thought from Ancient to Modern Times", Oxford University Press (1972) 1029, 1031, 1036. http://books.google.de/books?id=Lvco6V8fBpoC&pg=PA1029&lpg=PA1029&dq=%22gradually+ and+unwittingly+mathematicians+began+to+introduce%22&source=bl&ots=BmiyX56Ts_&sig=Jl H8x1AL88Q- 8DWDyoEpmMCJVdI&hl=de&sa=X&ei=Mr9uUIq0BeLl4QSIlIDQBw&sqi=2&ved=0CDMQ6AEwA g#v=onepage&q=%22gradually%20and%20unwittingly%20mathematicians%20began%20to%2 0introduce%22&f=false [2] H. Weyl: "Mathematics and logic", American Mathematical Monthly 53, 1946, p. 2. [3] H. Weyl: "Über die neue Grundlagenkrise der Mathematik", Math. Zeitschrift 10 (1921) reprinted in: "Gesammelte Abhandlungen, II", Springer, Berlin (1968) 149f. http://books.google.de/books?id=OGnthgn0H9AC&pg=PA143&hl=de&source=gbs_toc_r&cad=4 #v=onepage&q&f=false http://de.scribd.com/doc/49885193/Hermann-Weyl-Ueber-Die-Neue-Grundlagenkrise-Der- Mathematik

§ 114 The requirement that every element of a set shall be a set itself seems questionable. Formally that may work and simplifies the formalism. But what about the application of set theory on geometry and physics? {{That appears to be the least important problem. Set theory has no application anyway.}} [Zermelo to Fraenkel, Jan. 20, 1924] For German original texts see: Das Kalenderblatt 100319 http://www.hs-augsburg.de/~mueckenh/KB/KB%20201-400.pdf

§ 115 According to Russell, the structure of the infinite and the continuum were completely revealed by Cantor and Dedekind, and the concept of an infinitesimal had been found to be incoherent and was “banished from mathematics” through the work of Weierstrass and others [1901, pp. 88, 90]. These themes were reiterated in Russell’s often reprinted Mathematics and the Metaphysician [1918] and further developed in both editions of Russell’s The Principles of Mathematics [1903; 1937], the works which perhaps more than any other helped to promulgate these ideas among historians and philosophers of mathematics. In the two editions of the latter work, however, the banishment of infinitesimals that Russell spoke of in 1901 was given an apparent theoretical urgency. No longer was it simply that “nobody could discover what the infinitely little might be,” [1901, p. 90] but rather, according to Russell, the kinds of infinitesimals that had been of principal interest to mathematicians were shown to be either “mathematical fictions” whose existence would imply a contradiction [1903, p. 336; 1937, p. 336] or, outright “self-contradictory,” as in the case of an infinitesimal line segment [1903, p. 368; 1937, p. 368]. In support of these contentions Russell could cite no less an authority than Georg Cantor, the founder of the theory of infinite sets. Having accepted along with Russell that infinitesimals had indeed been shown to be incoherent, and that (with the possible exception of constructivist alternatives) the nature of the infinite and the continuum had been essentially laid bear by Cantor and Dedekind, following the development of in 1961, a good number of historians and philosophers of mathematics (as well as a number of mathematicians and logicians) readily embraced the now commonplace view that is typified by the following remarks:

In the nineteenth century infinitesimals were driven out of mathematics once and for all, or so it seemed. [P. Davis and R. Hersh 1972, p. 78]

But ... the German logician (1918–1974), who invented what is known as non- standard analysis, thereby eventually conferred sense on the notion of an infinitesimal greater than 0 but less than any finite number. [Moore 1990; 2001, p. 69]

Indeed ... nonstandard analysis ..., created by Abraham Robinson in the early 1960s, used techniques of mathematical logic and to introduce a rigorous theory of both [non-Cantorian] infinite and infinitesimal numbers. This, in turn, required a reevaluation of the long-standing opposition, historically, among mathematicians to infinitesimals in particular. [Dauben 1992a, pp. 113-114]

[Philip Ehrlich: "The Rise of non-Archimedean Mathematics and the Roots of a Misconception I: The Emergence of non-Archimedean Systems of Magnitudes", Arch. Hist. Exact Sci. 60 (2006) 1-121]

And the moral of this story? Not long ago there existed in mathematics a very questionable opinion of the past that finally has been overcome. That must not happen in matheology - the only science standing absolutely safe.

§ 116 How can the assumption of the infinite be justified? Could not just this seemingly so fruitful hypothsesis of the infinite have introduced straigth contradictions into mathematics, thereby destroying the basic nature of this science that is so proud upon its consistency? [On the hypothesis of the infinite, Ernst Zermelo's Warsaw notes W4 (p. 171), reported in H.-D. Ebbinghaus, V. Peckhaus: "Ernst Zermelo, An Approach to His Life and Work", Springer (2007) p. 292.] For German original texts see: Das Kalenderblatt 100322 http://www.hs-augsburg.de/~mueckenh/KB/KB%20201-400.pdf

§ 117 Finally, considering all the applications and recognizing the whole host of transfinite conclusions of the most difficult and painstaking sort that are involved for instance in relativity theory and quantum theory, and how nature precisely follows these results, the beam of the fixed star, Mercury, and the complicated spectra here on earth and in a distance of hundred thousands of lightyears. Should we in view of these facts hesitate only one second to apply tertium non datur {{to infinite sets}} only because of the beautiful eyes of Kronecker and because of some philosophers who are disguised as mathematicians and for reasons that are completely arbitrary and not even can be formulated precisely? (David Hilbert 1931, 387f) [Volker Peckhaus: "Becker und Zermelo"]

{{It seems that Hilbert really believed in the scientific application of transfinity. Concerning the rumor that at a time not more than twelve scholars had understood (general) relativity theory, we can safely say that Hilbert was not among that dozen.}} Under this aspect Zermelo's judgement (in a letter of August 28, 1928 to Marvin Farber) about Hilbert's (and Ackermann's) Foundations of Theoretical Logic becomes understandable: "Hilbert's just published logic is more than miserable, and also from his long announced Foundations of Mathematics I do no longer expect anything spectacular." [H.-D. Ebbinghaus, V. Peckhaus: "Ernst Zermelo, An Approach to His Life and Work", Springer (2007) 293]

Zermelo defended his point of view with clear insights and discerning arguments, but also with polemical formulations and sometimes hurtful sharpness. The controversial attitude shining through here has become a dominating facet of his image {{that makes him simpatico}}. Further controversies such as those with Ludwig Boltzmann about the foundations of the kinetic theory of heat and with Kurt Gödel and Thoralf Skolem about the finitary character of mathematical reasoning support this view. [loc. cit. p. VII, preface] http://books.google.de/books?id=G1nQU0GKvx8C&pg=PR7&lpg=PR7&dq=%22his+point+of+vi ew+with+clear+insights%22&source=bl&ots=TiX2_dD9rX&sig=jZYPErW5-Fg7eJXx9dGje2P- M9Q&hl=de&ei=Ei6iS5bNIKSAnQPmhMCFCg&sa=X&oi=book_result&ct=result&resnum=1&ved =0CAYQ6AEwAA#v=onepage&q=%22his%20point%20of%20view%20with%20clear%20insight s%22&f=false

For German original texts see: Das Kalenderblatt 100323 http://www.hs-augsburg.de/~mueckenh/KB/KB%20201-400.pdf

§ 118 Marvin Farber took "Notes on the foundations of Mathematical Logic" from discussions with Zermelo between 24 April and 10 June 1924 (8 page typescript). Models of axiom systems are conceived as substrates. A substrate is "a system of relations between the elements of a domain". It is here that Zermelo clearly distances himself from Hilbert's programme of consistency proofs. With regards to the still open question of the consistency of arithmetic it says (ibid., 2): A substrate is presupposed, as, for example, in the case of the series of real numbers. Something is presupposed which transcends the perceptual realm. The mathematicians must have the courage to do this, as Zermelo states it. In the assumption or postulation of a substrate, the freedom of contradiction of the axioms is presupposed [...]. Zermelo differs from Hilbert on this. It is Hilbert's view that it must be proved. The scientific estrangement from Hilbert my have led Zermelo to feel also a personal one. {{Finally however Zermelo has won: The freedom of contradiction of the axioms is presupposed and, as we can learn by an infinitude of striking examples, always postsupposed too.}} [H.-D. Ebbinghaus, V. Peckhaus: "Ernst Zermelo, An Approach to His Life and Work", Springer (2007) p. 156]

§ 119 The point of all this is that just as the finiteness of our physical bodies does not imply that every physical object is finite, the finiteness of the number of cells in our brains does not mean that every mental object is finite. Well ... are there any infinite minds, thoughts, ideas, or forms or what have you in the Mindscape? [...] If infinite forms are actually out there in the Mindscape, then maybe we can, by some strange trick of mental perspective, see some of these forms. {{One of these strange tricks is presumably the consumption of enough alcohol.}} [Rudy Rucker: "Infinity and the Mind", Princeton University Press, Princeton (2005) p. 38]

§ 120 f = x2 fl df/dx = 2x + dx

The infinitesimal dx disappears because it is much smaller than the finite 2x, explains Marquis de l'Hospital in the first textbook on Calculus. But what happens at x = 0 ?, asks D. Laugwitz in "Zahlen und Kontinuum" on p. 25.

§ 121 It is not unknown that mathematics besides the large benefit for the practical life has also a second, not less important although less obvious benefit for the practicing and sharpening of the intellect. The latter is what the state mainly is intending when prescribing the study of mathematics for every kind of university study. [B. Bolzano: "Betrachtungen über einige Gegenstände der Elementargeometrie“ (1804)]

{{Bolzano is the inventor of the word Menge (set). But since Menge in German means "a lot", he declared:}} Also allow me to call a collection, that contains only two parts, a set. [J. Berg (Hrsg.): "Bernard Bolzano, Einleitung zur Grössenlehre", Friedrich Frommann Verlag, Stuttgart (1975), Bolzano-Gesamtausgabe, Reihe II Band 7, p. 152].

A Menge consisting of only one or none element Bolzano would hardly have recognized as a Menge. When will the first sets of negative cardinality be invented? Or has it happened already?

For German original texts see: Das Kalenderblatt 091223 http://www.hs-augsburg.de/~mueckenh/KB/KB%20201-400.pdf

§ 122 The concept of infinity has been for hundreds of years one of the most fascinating and elusive ideas to tantalize the minds of scholars and lay people alike. The theory of infinite sets lies at the heart of much of mathematics, yet is has produced a series of paradoxes that have led many scholars to doubt the soundness of its foundations {{not the disciples of matheology, of course}}. [Naum Yakovlevich Vilenkin: "In search of infinity", Birkhäuser, Boston (1995), Cover] http://books.google.de/books?id=cU3HQFek7L0C&source=gbs_navlinks_s http://www.weltbild.de/3/16939032-1/buch/in-search-of-infinity.html

§ 123 Suppose a contradiction were to be found in the axioms of set theory. Do you seriously believe that a bridge would fall down? http://www-history.mcs.st-and.ac.uk/Quotations/Ramsey.html

§ 124 The third point is that under these conditions it is straightforward to show that the procedure “Give me any numeral n you can imagine, I will give you the next one” has to break down at a certain point. Ask any person to imagine a very large numeral, say, in decimal presentation. Usually what we do is to form a picture, say, we see a blackboard and it is covered with ciphers all over. But that won’t do. For once we have such a picture, it is obvious that is communicable, hence that it is finitely expressible and hence that there is room to imagine the next numeral and to communicate it. Thus, the alternative must be that the numeral is so large that it cannot be imagined, thereby making it senseless to talk about the next one. I will return to the implicit paradoxical nature of what I just wrote. What is being asked is to imagine a numeral so huge that it cannot be imagined. [Jean Paul Van Bendegem: "Why the largest number imaginable is still a finite number", p. 11]

(a) Labels are used merely as labels: if the world is finite, so is the set of labels, and it is impossible to label all “objects” in the world. (b) Labels form a structured set. In this case the labelling process can become more economical and more efficient, but it remains the case that the set of labels stays finite. (c) Labels form a structured set inserted in a theoretical framework. Here two subcases can be distinguished: (c1) There are interpretations of the theoretical framework that refer to “objects” in the world. Obviously in this case everything remains finite again on the assumption that the world is finite. (c2) There are no specific interpretations that refer to “objects” in the world. It is then always possible to find finite quasi-models that are derived from the classical infinite models of the theoretical framework. In some cases (as shown in the example above) these quasi-models can be seen as extensions of the classical model since it is possible to keep all classically true statements true in the quasi-model. Thus in those cases no truths are lost. The last case also applies to all labels that can be imagined by a labelling machine, if the requirement is that the labels should be communicable. Hence, if it is representable, it is obvious that we can imagine something larger, as we usually represent something in an environment, hence additional space is available. What we have to imagine, is a label such that if we try to represent it, we should fail to do so. Hence the agreement with ’s description quoted at the beginning of this paper: “so large that it has no physical or psychological significance …”. It is paradoxical to be sure. If formulated in terms of questions, the problem becomes immediately obvious. The question “What is the largest label or numeral that is not imaginable?”, should not be answered by “The label so-and-so with properties such-and-uch”, because then it has been imagined, thereby not answering the question. The answer must be: “Whatever it is, that label”. An alternative reply would be: “The largest label is that label about which questions such as the question posed cannot be asked”. It is that label that ceases to be that label as soon as something is said about it. {{A property that can change - potential infinity.}} A conclusion that fits in nicely with the argued for vagueness of the largest label. [Jean Paul Van Bendegem: "Why the largest number imaginable is still a finite number", p. 16] http://www.vub.ac.be/CLWF/members/jean/the%20largest%20number%20imaginable.pdf

Consider a label that has been constructed using all matter of the universe except that little heap that is necessary to maintain your consciousness. If you try to increase the Kolmogorov- complexity of the label, then you will have to use matter from said heap. This process must stop somewhere. But you will no longer be able to recognize its limit.

§ 125 The leap into the beyond occurs when the sequence of numbers that is never complete but remains open toward the infinite is made into a closed aggregate of objects existing in themselves. Giving the numbers the status of objects becomes dangerous only when this is done. [p. 38] In advancing to higher and more general theories the inapplicability of the simple laws of classical logic eventually results in an almost unbearable awkwardness. And the mathematician watches with the larger part of his towering edifice which he believed to be built of concrete blocks dissolve into mist before his eyes. [p. 54] [Hermann Weyl: "Philosophy of Mathematics and Natural Science" (1949); reprinted with a new introduction by Frank Wilczek, Princeton Univ. Press (2009)] http://press.princeton.edu/titles/8960.html http://frankwilczek.com/weyl05.pdf

§ 126 (A) Mathematics is common sense; (B) Do not ask whether a statement is true until you know what it means; (C) A proof is any completely convincing argument; (D) Meaningful distinctions deserve to be preserved. [Errett Bishop: "Schizophrenia in contemporary mathematics", Amer. Math. Soc. Colloquium Lecture, Seventy-eighth summer meeting, University of Montana, Missoula, Montana (1973)] Mathematical Reviews (MathSciNet): MR788163 http://www-history.mcs.st-and.ac.uk/Biographies/Bishop.html http://en.wikipedia.org/wiki/Errett_Bishop

§ 127 The intercession is an alternative measure for infinite sets of finite numbers. Definition: Two infinite sets, A and B, intercede (each other) if they can be put in an intercession, i.e., if they can be ordered such that A is dense in B and B is dense in A. In other words, between two elements of A there is at least one element of B and, vice versa, between two elements of B there is at least one element of A. The intercession of sets with nonempty intersection, e.g., the intercession of a set with itself, requires distinction of identical elements. As an example an intercession of the set of positive integers and the set of even positive integers, 1, 2, 3, ... and 2', 4', 6', ..., is given by 1, 2', 2, 4', 3, 6', .... The intercession includes Cantor's definition of equivalent (or equipotent) sets by one-to-one correspondence (or bijection): Two equivalent sets always intercede each other, i.e., they can always be put in an intercession. The intercession is an equivalence relation. All infinite sets of finite numbers (like the integers, the rationals or the reals) belong, under this relation, to the same equivalence class. [W. Mückenheim: "Die Mathematik des Unendlichen", Shaker, Aachen (2006) 116-117] http://planetmath.org/encyclopedia/Intercede.html

§ 128 Robert Grosseteste (1168 - 1253), Bishop of Lincoln and teacher of Roger Bacon in Oxford claimed: "The number of points in a segment one ell long is its true measure." Also John Baconthorpe (? - 1346), called Doctor resolutus, brought honour on his epithet and courageoulsly opposed the contemporary scholastic opinion "infinitum actu non datur" by stating: "There is the actual infinite in number, time, quantity."

§ 129 Unfortunately, I was introduced to Cantor in a course of . To say that most mathematics professors who teach Cantor's ideas are mentally challenged, is being rather kind to them. My first impressions of Cantor were not good, but my impressions of the professors who still teach his theories, are significantly worse. (John Gabriel) Vox populi - vox Dei ?

§ 130 What is so bad about contradictions? I shall address the title question, and the answer I shall give is: rnaybe nothing much. Let me first explain how, exactly, the question is to be understood. I shall Interpret it to mean: What is wrong with believing some contradictions? I emphasize the 'some'; the question 'What is wrong with believing all contradictions?' is quite different, and, I am sure, has a quite different answer. It would be irrational to believe that I am a fried egg. [...]. I think that there is nothing wrong with believing some contradictions. I believe, for example, that it is rational (rationally possible - indeed, rationally obligatory) to believe that the liar sentence is both true and false. [...] I have discovered, in advocating views such as this, that audiences suppose them to be a priori unacceptable. When pressed as to why, they come up with a number of arguments. I shall consider five of the most important, and show their lack of substance. They can be summarized as follows: (1) Contradictions entail everything. (2) Contradictions cannot be true. (3) Contradictions cannot be believed rationally. (4) If contradictions were acceptable, people could never be rationally criticized. (5) If contradictions were acceptable, no one could deny anything. I am sure that there must be other possible objections, but the above are the most fundamental that I have encountered. I shall take them in that order. What I have to say about the first objection is the largest, because it lays the basis for all the others. [...] The objection is that rational belief is closed under entailment, but a contradiction entails everything. Hence, if someone believed a contradiction, they ought to believe everything, which is too much. I certainly agree that believing everything is too much: I have already said that there is an important difference between some and all. {{The following contradiction certainly does not belong to the category "some: "The absence of an object in the union of all finite initial sequences implies its absence in the infinite set." "The absence of an object in the union of all finite initial sequences does not imply its absence in the infinite set."}} [: "What is so bad about contradictions?", JSTOR The Journal of Philosophy, Vol. 95, No. 8 (Aug. 1998) 410-426] http://www.jstor.org/pss/2564636

§ 131 The following correspondence has been made available to me by one of the correspondents.

Dear NN, [...] The key to Cantor's argument (as it is to Dedekind's definition of the reals via Dedekind cuts) is the notion of a completed infinity - anathema to Aristotle and much later to Gauss, but the rock on which all of our modern view of set theory - and hence mathematics {{how can a rational thinking human make such an error?}} - is based. So in Cantor's proof, we have to take the purported list of all positive reals in (0, 1), in 1-1 correspondence with the naturals, as completed before we look further. There is no room to add another, nor any changes that can be made: all the positive reals in (0, 1) are already enumerated. Now begin his diagonalization: as the diagonal passes through the nth entry, we make the 5-6 change in 'our' decimal in accordance with what we find in the nth place of the nth entry. Should what we have so far made occur later in the list, as the rth entry, our diagonal line will pass though it too, and we'll make the same kind of change in its rth decimal place. No conundrum after all, then - if you're willing to accept the idea of a completed infinity (here in the form of a completed list) with all of its implications. The constructivists aren't willing so to accept - but this leaves them with problems perhaps even more serious (such as being unable to prove the intermediate value theorem) {{this situation is certainly not improved by accepting a contradiction}}. You might conclude that all's not well in this mathematical Denmark - perhaps we need to rethink the foundations on a basis other than set theory, if it leads to such outrageous consequences. {{The foundations are already there. Please do not go astray in the infinite. You will not find them there. Foundations are always on the bottom: I + I = II , I + II = III , ... }}

§ 132 The following correspondence has been made available to me by one of the correspondents.

The reason that I did not answer your email message earlier is that I was re-reading Mueckenheim's papers to understand his arguments better. Somehow I do not interpret what Mueckenheim says as you did. I think you are looking at Mueckenheim's work through the eyes of a constructivist and Mueckenheim is not a constructivist. There is, I believe, a deeper message in Mueckenheim's work that we are applying a different standard of interpretation or reasoning to Cantor's proof of uncountability of reals than his proof of countability of rationals. {{That has been clearly recognized by the writer: For the "proof of countability" the finite representations are used, for the "proof of uncountability" the infinite representations are used, which do not exist without finite definitions and which, therefore, cannot be counted at all.}} I don't know how to explain it but there are some hidden assumptions about the list of reals that Cantor based his diagonal argument on. For example, Cantor used the idea completed infinity here, meaning he assumed all reals were in the list! But he said nothing about how the list was constructed! In fact, if I give you a real number, you could not tell me where the number is in the list. Accepting the assumption of completed infinity is, in my mind, equivalent to accepting God on faith! I am suddenly feeling a connection between religion and math. {{That is not surprising with respect to Cantor's world view, and it is not reprehensible either - but it is not related to mathematics.}} Basically, if you can construct the list of reals, it will automatically be countable! {{Of course. We can only construct what can exist in principle.}} So Cantor started with a contradiction right from the beginning of his diagonal argument.

§ 133 Each real number of the interval [0, 1] can be represented by an infinite path in a given binary tree. In Section 2 the binary tree is projected on a grid N × N and it is shown that the set of the infinite paths corresponds one-to-one to the set N. The Theorems 2.1 and 2.2 give the first proof and the Theorem 2.3 provides a second proof. Section 3 examines the Cantor’s proof of 1891. The Section shows that (i) if the diagonal method is correct, then any denumerable list L to which the diagonal method was applied is incomplete (Theorem 3.1), (ii) if some complete list exists and if the diagonal method is correct, then a complete list cannot be represented in the form used in Cantor’s proof (Theorem 3.2) and (iii) being L incomplete nothing affirms or denies that |N| is the cardinality of the set of real numbers of the interval [0, 1] (Theorem 3.3). Sections 2 and 3 mean that (i) there is the list, denoted by LH, such that it contains all members of N and all members of F and to each member of F of the list corresponds one-to-one a member of N of the list and (ii) if the diagonal method used in Cantor’s 1891 proof [1] is valid, then LH has a different form of the form of the list L used by Cantor. In Section 4 we try to show that LH has the same form as L. [J. C. Ferreira: "The cardinality of the set of real numbers" (2001)] http://arxiv.org/PS_cache/math/pdf/0108/0108119v6.pdf

§ 134 The sentence "This irrational number exists" means 1) This irrational number has a name; and 2) we can decide whether it is less than or greater than any rational number we might name.

Our idea of length is that it is continuous. And since we think of measuring length as the distance from the origin O along the x-axis, the thought was that the values of x must reflect the continuum of lengths by being a continuum of numbers. One way to express that is to say that, corresponding to each endpoint P of OP, there is a real number x, the coordinate of P, which is the measured length of OP. In other words, we must be able to measure every length. But will that be possible? Will it be possible to name the ratio that every length will have to a unit of measure? No. It is impossible to name every point in a continuum – a continuum of names is an absurdity. Names are discrete. And nameless numbers do not exist, not even potentially. There is no arithmetical continuum. [...]

Infinite decimals? "Alice laughed: "There's no use trying," she said; "one can't believe impossible things." "I daresay you haven't had much practice," said the Queen. "When I was younger, I always did it for half an hour a day. Why, sometimes I've believed as many as six impossible things before breakfast." (Alice in Wonderland)

Each real number in the supposed continuum between 0 and 1, at any rate, has that form. But an infinite decimal has no name. It is not that we will never finish naming it. We cannot even begin. Infinite decimals, therefore, since they do not have names, are not numbers. Just because something is written with the symbols we use for numbers -- 1, 2, 3, and so on -- does not make it a number, any more than something written with the symbols we use for words -- "obakqe" -- makes it a word.

Equivalently, infinite decimals are not numbers because with them it is impossible to solve the four problems of arithmetic. We cannot name the sum of infinite decimals; we cannot name their difference; we cannot name their product; and we cannot name their quotient. Infinite decimals are not numbers.

If a student in an arithmetic class were to say, "Although I cannot name it, teacher, the sum exists; and to know that is sufficient," then the student might deserve an A in metaphysics, but in arithmetic she would certainly fail.

The symbol for an infinite decimal, although it is called a real number, is intended to refer to a point on what is called the real line. But a point is a concept completely different from a number. And to achieve their identity by postulation is both a tautology -- "To every point on the line there corresponds a point on the line" -- and an acknowledgement of defeat.

We can try to make sense of an infinite decimal, however, as being an abbreviation for a limit. [...] To suppose, however, that there could be algorithms for computing a continuum of real numbers, would require a continuum of algorithms. Again that is absurd. Algorithms are discrete. And in the absence of an algorithm, it will be impossible to place a supposedly infinite decimal, such as .24059165378..., with respect to order relative to any rational number.

In the absence of an algorithm, .24059165378... is nothing but a sequence of made up digits followed by three dots. It is not the symbol of a number. In fact, the English mathematician and father of artificial intelligence Alan Turing proved the following: To compute the decimal expansion of a real number, it is possible to create an algorithm for only a countable number of them. Why the obsession with a continuum of numbers? It was aggravated by the apparent demand of coordinate geometry: For every point on the x-axis there must be a number which is its coordinate. But it does not matter that an arithmetical continuum is a fiction. In the actual practice of calculus, it never comes up. When we do a calculation, we name a number. That is all anyone has ever done or ever will do, even though the theoretical explanation for what we do might be nonsense. {{What sense would it make to maintain that obvious nonsense?}}

In short, inasmuch as measurements -- numbers that we can know and name -- are the of the physical sciences, the theory of real numbers is not a theory of measurement. Together with its associated set theory ("The set of real numbers", "The set of points on a line"), the theory of real numbers is the most prominent current example of fantasy mathematics.

Lawrence Spector (2010) For a newer version look here: http://www.themathpage.com/aCalc/real.htm

§ 135 One central agent of the connection between mathematics and religion is the concept of infinity (but it is not the only one!). From its first appearance under the name of "apeiron" with Anaximander of Miletus (610-546 BC), to the recent work of Hugh Woodin [9], this is a permanent theme in mathematics -- H. Weyl even wrote that mathematics is "the science of the Infinite" [8] -- but the theme is also permanent in the philosophy of mathematics, and the word End is not yet written. This is a fascinating story that has inspired philosophers and theologians, poets and mathematicians. One can follow the birth of the concept, corresponding to attributes of God (or space or time) with mathematics filling more and more space through the centuries, until the Cantorian parthenogenesis between mathematics and religion (but still with a trace of its origins with the theological absolute to escape the paradox of the set of all sets). [...] A second theme running through many chapters of the book is the search for a global vision uniting mathematics and religion. This can be found first in the school of Pythagoras, the object of "The Pythagoreans," an interesting study by Reviel Netz. Netz suggests, through an analysis of the mystery of the Pythagorean cult, that religion and mathematics might be able to interact, because they share some way of "rationalizing mystery" through analogies and metaphors. The global unity of mathematics with religion is central in Plato's work, and in his followers' such as and , but also much later in modern times. [8] Hermann Weyl, The Open World (God and the Universe, , Infinity), Yale, 1933, Reprint Oxbow Press, 1989. [9] Hugh W. Woodin, "The continuum hypothesis. I.," Notices Amer. Math. Soc.48 (2001) 567- 576. Part II. Notices Amer. Math. Soc. 48 (2001) 681-690. [Mathematics and the Divine. A Historical Study edited by Teun Koetsier and Luc Bergmans, Amsterdam, , 2005, Hardbound, 716 pp., US $250, ISBN-$3: 978-0-444-50328-2, ISBN- IO: 0-444-50328-5 Rewieved by Jean-Michel Kantor in The Mathematical Intelligencer 30, 4 (2008) 70-71]

§ 136 By Cantor was - it is well known - as biggest beast alephant grown. But aleph is a number that turned out too large - thus ant fell dead. under poetic licence. After Christian Morgenstern: http://www.gedichtsuche.de/gedichtanfaenge_ueber.php?id=278. http://en.wikipedia.org/wiki/Christian_Morgenstern

§ 137 A moment of contemplation, of resting, of reflection: When have you identified a number by an infinite string of symbols for the last time? Was there a first time?

§ 138 "God himself cannot persist without wise men" – Luther said, and with every right; but "God himself can even less persist without unwise men" – that good Luther did not say! [F. Nietzsche: "Die fröhliche Wissenschaft", 3. Buch, Schmeitzner, Chemnitz (1882)] http://gutenberg.spiegel.de/buch/3245/6 {{By the way, same holds for matheology.}}

In this way an ambitious innovator always attains his goal; he becomes a famous philosopher and the corruption of youth happens on a large scale. [Cantor to Loofs, Feb. 24, 1900, about Nietzsche]

Corruptor of youth? Isn't Kronecker quoted with just those words? For instance by Manin: [Yuri I. Manin:Georg Cantor and his heritage, arXiv:math/0209244v1] http://aps.arxiv.org/PS_cache/math/pdf/0209/0209244v1.pdf

§ 139 The question of greatest urgency confronting nineteenth- and early twentieth-century mathematicians was arguably that of the status of the infinite within mathematics. Zermelo’s s1921, comprising five multi-part philosophical “theses”, should be understood in that spirit. Despite its brevity, s1921 is somewhat repetitive. It seems that Zermelo had no of publishing it even as part of some longer piece. Instead, s1921 likely functioned as a personal manifesto; clearly, Zermelo sees himself as breaking new ground here {{breaking new ground? - or breaking through thin ice?}}. If conceived in July 1921, in fact, s1921 would contain the earliest intimation of the theory of systems of infinitely long propositions [...] [R. Gregory Taylor: "Introductory note to 'Zermelo s1921 - Theses concerning the infinite in mathematics'", Ernst Zermelo - Collected Works/Gesammelte Werke Volume I - Set Theory, Miscellanea / Band I - Mengenlehre, Varia (Schriften der Mathematisch-naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften, 2010, Volume 21, 302-307)] http://www.springerlink.com/content/w4301h8878h711t6/

§ 140 Ms C dies and goes to hell, or to a place that seems like hell. The devil approaches and offers to play a game of chance. If she wins, she can go to heaven. If she loses, she will stay in hell forever; there is no second chance to play the game. If Ms C plays today, she has a 1/2 chance of winning. Tomorrow the probability will be 2/3. Then 3/4, 4/5, 5/6, etc., with no end to the series. Thus every passing day increases her chances of winning. At what point should she play the game? The answer is not obvious: after any given number of days spent waiting, it will still be possible to improve her chances by waiting yet another day. And any increase in the probability of winning a game with infinite stakes has an infinite utility. For example, if she waits a year, her probability of winning the game would be approximately .997268; if she waits one more day, the probability would increase to .997275, a difference of only .000007. Yet, even .000007 multiplied by infinity is infinite. On the other hand, it seems reasonable to suppose the cost of delaying for a day to be finite - a day's more suffering in hell. So the infinite expected benefit from a delay will always exceed the cost. This logic might suggest that Ms C should wait forever, but clearly such a strategy would be self defeating: why should she stay forever in a place in order to increase her chances of leaving it? So the question remains: what should Ms C do?' [E.J. Gracely: "Playing games with eternity: The devil's offer", Analysis 48.3 (1988) p. 113] http://www.balliol.ox.ac.uk/sites/default/files/Dudman-1988-Indicative-and-Subjunctive.pdf {{The verdict of eternal damnation - one of few practical applications of set theory.}}

§ 141 The initially warm relationship between Hilbert and Brouwer began to cool in the twenties, when Brouwer started to campaign for his foundational views. Hilbert accepted the challenge - he took the threat of an intuitionistic revolution seriously. Brouwer lectured successfully at meetings of the German Mathematical Society. His series of Berlin lectures in 1927 caused a considerable stir; there was even some popular reference to a Putsch in mathematics. [Dirk van Dalen: "The War of the Frogs and the Mice", The Mathematical Intelligencer 12, 4 (1990) 17-31]

Brouwer came to Göttingen to deliver a talk on his ideas to the Mathematics Club. "You say that we can't know whether in the decimal representation of π ten 9's occur in succession," someone objected after Brouwer finished. "Maybe we can't know - but God knows!" {{Isn't "matheology" an appropriate label?}} To this Brouwer replied dryly, "I do not have a pipeline to God." After a lively discussion Hilbert finally stood up. "With your methods," he said to Brouwer, "most of the results of modern mathematics would have to be abandoned {{compare Franz Lemmermeyer's accusation that I am at war with modern mathematics, abandoning the results oft the last 2500 years http://www.zentralblatt-math.org/zmath/en/search/?q=an:1204.00016&format=complete http://www.hs-augsburg.de/~mueckenh/Kommentar/ }}, and to me the important thing is not to get fewer results but to get more results." {{Even on the risk that the "results" are of as little value as the results that can be derived from the bodily Assumption of Virgin Mary?}} He sat down to the enthusiastic applause. {{Four legs good, two legs bad.}} [Constance Reid: "Hilbert", Springer (1970) p. 184f] http://books.google.de/books/about/Hilbert.html?id=mR4SdJGD7tEC&redir_esc=y

§ 142 Baire considered any infinite set, denumerable or not, as ‘virtual’ - an object defined by certain conventions. Thus if one is given an infinite set, “it is false ... to consider the subsets of this set as given”. {{Baires point of view at first glance seems unresonable and disconcerting. How can it be that the subsets of a given set can be not given if even an axiom "proves" that the power set is given? But Baire has received a late confirmation. It has turned out that some elements of some sets of real numbers and some elements of the power set of the natural numbers cannot be identified by finite definitions whereas infinite sequences without finite definition cannot be identified at all. Thus they are not given, at least they cannot be taken.}} A fortiori it made no sense to conceive, as Zermelo did, that the choice of an element had been made in each subset. {{That is plainly impossible if the objects are not given.}} [...] Baire insisted on regarding Zermelo’s as dependent: “One takes a distinguished element m1 from the set M; there remains M - m1, in which one takes a distinguished element m2, etc” {{Zermelo felt obliged to show that γ-sets with one and two elements exist. This mathematically completely superfluous action is the psychologically important abracadabra of the magician. So Zermelo elegantly passes by the proof that every subset of a given set exists. Had he tried it, his failure would have become obvious immediately. But better late than never!}} [Gregory H. Moore: "The Origins of Zermelos Axiomatization of Set Theory" (1978)] http://www.jstor.org/pss/30226178

§ 143 This article undertakes a critical reappraisal of arguments in support of Cantor’s theory of transfinite numbers. The following results are reported: - Cantor’s proofs of nondenumerability are refuted by analyzing the logical inconsistencies in implementation of the reductio method of proof and by identifying errors. Particular attention is given to the diagonalization argument and to the interpretation of the axiom of infinity. - Three constructive proofs have been designed that support the denumerability of the power set of the natural numbers, P(Ù), thus implying the denumerability of the set of the real numbers —. These results lead to a Theorem of the Continuum that supersedes Cantor’s Continuum Hypothesis and establishes the countable nature of the real number line, suggesting that all infinite sets are denumerable. Some immediate implications of denumerability are discussed: - Valid proofs should not include inconceivable statements, defined as statements that can be found to be false and always lead to contradiction. This is formalized in a Principle of Conceivable Proof. - Substantial simplification of the axiomatic principles of set theory can be achieved by excluding transfinite numbers. To facilitate the comparison of sets, infinite as well as finite, the concept of relative cardinality is introduced. - Proofs of incompleteness that use diagonal arguments (e.g. those used in Gödel’s Theorems) are refuted. A constructive proof, based on the denumerability of P(Ù), is presented to demonstrate the existence of a theory of first-order arithmetic that is consistent, sound, negation- complete, decidable and (assumed p.r. adequate) able to prove its own consistency. Such a result reinstates Hilbert’s Programme and brings arithmetic completeness to the forefront of mathematics. [J. A. Perez: "Addressing mathematical inconsistency: Cantor and Gödel refuted" Arxiv (2010)] http://arxiv.org/ftp/arxiv/papers/1002/1002.4433.pdf

§ 144 Introduction. I dedicate this essay to the two-dozen-odd people whose refutations of Cantor’s diagonal argument (I mean the one proving that the set of real numbers and the set of natural numbers have different cardinalities) have come to me either as referee or as editor in the last twenty years or so. [...] Cantor’s argument is short and lucid. It has been around now for over a hundred years. Probably every professional mathematician alive today has studied it and found no fallacy in it.

{{Paul Bernays is no longer alive but his recognition is so much a commonplace that every living mathematician should know it: "It is not an exaggeration to say that platonism reigns today in mathematics. But on the other hand, we see that this tendency has been criticized in principle since its first appearance and has given rise to many discussions. This criticism was reinforced by the paradoxes discovered in set theory, even though these antinomies refute only extreme platonism. It is this absolute platonism which has been shown untenable by the antinomies. Nonetheless, if we pursue the thought that each real number is defined by an arithmetical law the idea of the totality of real numbers is no longer indispensable." [Paul Bernays: "On Platonism in Mathematics", (1934) p. 6f] And with extreme platonism also Cantor's diagonal argument vanishes. Who does know this and who does, in addition, share the scepticism of Solomon Feferman ["Infinity in Mathematics: Is Cantor Necessary?"]: "I am convinced that the platonism which underlies Cantorian set theory is utterly unsatisfactory as a philosophy of our subject [...] platonism is the medieval metaphysics of mathematics; surely we can do better." cannot share the absolute claim articulated here. The author should know it, and if knowing it, he should say it. But he says something quite different:}}

There is a point of culture here. Several of the authors said that they had trained as philosophers, and I suspect that in fact most of them had. In English-speaking philosophy (and much European philosophy too) you are taught not to take anything on trust, particularly if it seems obvious and undeniable. You are also taught to criticise anything said by earlier philosophers. Mathematics is not like that; one has to accept some facts as given and not up for argument. {{Philosophers often are charged to be bad mathematicians. Kant, for instance, according to Cantor [letter to Russell, Sept. 19, 1911], "was so bad a mathematician".}}

Nobody should be surprised when philosophers who move into another area take their habits with them. (In the days when I taught philosophy {{... that explains a lot!}}

[Wilfrid Hodges: "An editor recalls some hopeless papers", The Bulletin of Symbolic Logic 4,1 (1998)]

§ 145 This book advocates nothing less than the elimination of the infinite from mathematics. [...] If we accept Brouwer's view, the only sets which exist are those which are countable and have been effectively well-ordered. The author remarks, and this is his principal point, that we can as well go the whole way and admit the existence only of finite sets. Any statement about a countable, effectively well-ordered set can by a circumlocution be translated into a statement about the rule by which the elements follow each other in the well-ordering, which rule is something finite and definite. For example, we are shown how it is possible in his scheme to prove that every bounded monotonic sequence of irrational numbers has a limit. [...] It seems to me that the author, besides producing an interesting book, has made a good case for the contention that if we accept Brouwerism, we can get along theoretically without the notion of an infinite set, whether or not that notion is meaningless, as the author maintains. However, I do not believe that the views of Brouwer will ever find general acceptance among mathematicians. As this is not the place for an elaborate discussion of the questions raised by the intuitionists, I should merely like to add the following minor point to the prevailing confusion. If the continuum hypothesis is true, it is conceivable that someone may someday discover an effective way of well-ordering the real number continuum so that every number has only a countable number of predecessors. If this were done it would be practically a refutation of Brouwerism. It might seem, then, that either the intuitionists must prove that it cannot be done, or must proceed with a sort of sword of Damocles hanging over their heads. {{There seems to prevail a fundamental misunderstanding in the examiner's imagination. - Every real well-ordering would show the countability of the continuum, because then every real number would be named. But there are only countably many names. This paragraph shows that in 1931 the impossibility of a well- ordering of the continuum had not yet been generally recognized by mathematicians. Probably not much has changed yet, and students of mathematics don't even know what they are supposed to defend (see § 002).}} [Orrin Frink: Review of Felix Kaufmann: "Das Unendliche in der Mathematik und seine Ausschaltung", Deuticke, Leipzig (1930), Bull. Amer. Math. Soc. 37 (1931) 149-150.] http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/118 3494611

§ 146 Men who cannot work receive a pension. That's a commandment of humanity. Men who are dead cannot work. That's a hardship of misfortune. Men who are dead receive a pension. That's called a . Syllogism is a part of logic. Logic has been invented by Greeks. So, if in Greece someone is dead ... (Not every nonsense can be blamed to actual infinity. But most.)

§ 147 Every measurable in our reality has a Planck value that is a combination of three fundamental constants: c (the velocity of light), g (the gravitational constant), and h (Planck's constant). Planck values for time, length, area, volume, mass and any other measurable aspect of reality limit the maximal precision by which meaningful measurements may be taken. One Planck length is about 1.6ÿ10-35 m; the shortest interval of time is about 10-43 sec; volume cannot be broken into pieces smaller than about 10-99 cm3 [...] Planck‘s limits have some uncertainty, and the values may be refined in time, but what is clear is that real numbers applied to our reality are granular, i.e., they are discrete. All reals are counting numbers at the quantum level. [B. L. Crissey: "Unreal Irrationals: Turing Halts Cantor"] http://www.briancrissey.info/Research/Resume_files/Unreal%20Irrationals.pdf

§ 148 A one-to-one mapping (i.e. a bijection) between natural numbers, and real numbers between 0 & 1, is constructed; the mapping formula is simple, direct, and easy to calculate and work with. The traditional Cantor diagonal argument is then traced through: but at each step we use the mapping formula to show that the number generated thus far, is present in our one-to- one mapping; thus contradicting the traditional conclusion of said diagonal argument. We then extend the range of the mapping to the full set of real numbers, using the traditional tangent- function approach. The existence of the bijection naturally appears to contradict results in Cantor’s analysis. The present author speculates - but doesn’t pursue or prove here - that the apparent contradiction may be due to how the mappings here and in Cantor’s analysis, are constructed; and may be analogous to phenomena seen in rates of convergence of, &/or in naive rearrangements of conditionally-convergent, infinite series. [Edward Grattan: "A One-to-One Mapping from the Natural Numbers to the Real Numbers" (2012)] http://www.cruziero.com/conglom/11nr-vcurr.pdf

§ 149 It might be objected that no contradiction results from taking the real numbers to form a definite totality. There is, however, no ground to suppose that treating an indefinitely extensible concept as a definite one will always lead to inconsistency; it may merely lead to our supposing ourselves to have a definite idea when we do not ... [M. Dummett: "The seas of language", Oxford University Press (1993) p. 442] http://books.google.de/books/about/The_Seas_of_Language.html?id=dzf_nY- wI5IC&redir_esc=y

§ 150 There are many mathematicians who will accept the Garden of Eden, i.e. the theory of functions as developed in the 19th century, but will, if not reject, at least put aside the theory of transfinite numbers, on the grounds that it is not needed for analysis. {{In reality nearly all mathematicians do so.}} Of course, on such grounds, one might also ask what analysis is needed for; and if the answer is basic physics, one might then ask what that is needed for. When it comes down to putting food in one’s mouth, the 'need' for any real mathematics becomes somewhat tenuous. Cantor started us on an intellectual journey. One can peel off at any point; but no one should make a virtue of doing so. {{Neither of doing not so.}} [W.W. Tait: "Cantor’s Grundlagen and the " (2000) p. 21f] http://home.uchicago.edu/~wwtx/cantor.pdf

§ 151 The (truly) infinite, I claim, can never be subjugated. Indeed I would go further: the (truly) infinite, as a unitary object of thought, does not and cannot exist. This is not to say that the concept of the infinite has no legitimate use. One such use, if I am right, is precisely to claim that the infinite does not exist. Another such use, I would further argue, is to claim that there are infinitely many possibilities (including endlessly recurring possibilities of set membership) afforded by all the finite things that do exist. And to claim these things is, in a suitably neoteric way, to repudiate the actual infinite and to acknowledge the potential infinite - the very thing that Aristotle was teaching us to do some two and a half millennia ago. {{This view is not neoteric but simply sensible, rational and mathematical.}} [A.W. Moore: “The Infinite”, 2nd ed., Routledge, New York (2005) p. XV]

§ 152 Consider the following sequence of decimal numbers, consisting of digits 0 and 1

01. 0.1 010.1 01.01 0101.01 010.101 01010.101 0101.0101 ... which, when indexed by natural numbers, looks like this:

0211. 02.11 041302.11 0413.0211 06150413.0211 061504.130211 0817061504.130211 08170615.04130211 ...

What is the limit of the sequence of the sets of indexes on the left hand side? What is the limit of the decimal numbers?

§ 153 A charismatic speaker well-known for his clarity and wit, he once delivered a lecture giving an account of Gödel's second incompleteness theorem, employing only words of one syllable. [George Boolos: "Gödel's Second Incompleteness Theorem - Explained in Words of One Syllable", Mind, 103, Jan. 1994, p. 1ff] http://www2.kenyon.edu/Depts/Math/Milnikel/boolos-godel.pdf At the end of his viva, asked him, "And tell us, Mr. Boolos, what does the have to do with the real world?" Without hesitating Boolos replied, "It's part of it". {{If present at that illustrious moment I would have added another question: And tell us, Mr. Boolos, does every part of the real world have to observe its constraints? Unfortunately we don't know the answer. But the constraints of the speech would have been kept by a simple "yes".}} http://en.wikipedia.org/wiki/George_Boolos

The talk ended: So, if math is not a lot of bunk, then, though it can't be proved that two plus two is five, it can't be proved that it can't be proved that two plus two is five. By the way, in case you'd like to know: yes, it can be proved that if it can be proved that it can't be proved that two plus two is five, then it can be proved that two plus two is five. {{Spelled out clearly: If math is not a lot of bunk, then math is a lot of bunk. And this obvious nonsense not only has been accepted in matheology, but is sacred as a touchstone of the intellectual capacity of their disciples and as a fixing of their belief in finished infinity. - Because, as Gödel himself already noted, without actual infinity his theorems are invalid.}}

§ 154 Consistency Proof!

The long missed solution of an outstanding problem came from a completely unexpected side: Social science proves the consistency of matheology by carrying out a poll.

As reported in § 152 mathematics and matheology lead to different values of the

100 +101 10 +102 10 +103 1/10 += ... 0 (Cauchy) 10

100 +101 10 +102 10 +103 1/10 +> ... 1 (Cantor) 10

But 100 % of all matheologians who responded to our poll said that this difference is not surprising since different methods have been applied, namely the mathematical calculation invented by Cauchy and the matheological method invented by Cantor. Although both names begin with a C (like certainty (and even with a Ca (like can and cannot))) the following letters are completely different.

The general opinion is that it is not surprising to find different results when applying different methods. Even the application of the same method by different people may yield different results as we see daily in our elementary schools.

This attitude implies some consequences with respect to the human rights. We should no longer talk of mistakes and errors in calculations and punish pupils who deviate from the majority or main stream, but we should only note beside the result who applied what method and possibly also location and time because experience shows that the result of a calculation may depend on such details.

For, he reasons pointedly: That which must not, can not be. (C. Morgenstern)

§ 155 At first it seems obvious, but the more you think about it, the stranger the deductions from this axiom seem to become; in the end you cease to understand what is meant by it. (Bertrand Russell about the Axiom of Choice) [Naum Yakovlevich Vilenkin: "In search of infinity", Birkhäuser, Boston (1995) p. 123] http://books.google.de/books?id=cU3HQFek7L0C&printsec=frontcover&source=gbs_v2_summa ry_r&cad=0#v=onepage&q=&f=false

The axiom of choice is obvious. But there are no uncountable sets. Therefore the impossible task vanishes that elements must be well-orderable without the possibility to distinguish and identify them.

§ 156 The object of this paper is to give a satisfactory account of the Foundations of Mathematics in accordance with the general method of Frege, Whitehead and Russell. Following these authorities, I hold that mathematics is part of logic, and so belong to what may be called the logical school as opposed to the formalist and intuitionist schools. I have therefore taken Principia Mathematica as a basis for discussion and amendment; and believe myself to have discovered how, by using the work of Mr. Ludwig Wittgenstein, it can be rendered free from the serious objections which have caused its rejection by the majority of German authorities, who have deserted altogether its line of approach.

In this chapter we shall be concerned with the general nature of pure mathematics, and how it is distinguished from other sciences. (Footnote: In the future by 'mathematics' will always be meant 'pure mathematics'.) Here there are really two distinct categories of things of which an account must be given -- the ideas or concepts of mathematics, and the propositions of mathematics. This distinction is neither artificial nor unnecessary, for the great majority of writers on the subject have concentrated their attention on the explanation of one or other of these categories, and erroneously supposed that a satisfactory explanation of the other would immediately follow. Thus the formalist school, of whom the most eminent representative is now Hilbert, have concentrated on the propositions of mathematics, such as '2 + 2 = 4'. They have pronounced these to be meaningless formulae to be manipulated according to certain arbitrary rules, and they hold that mathematical knowledge consists in knowing what formulae can be derived from what others consistently with the rules. Such being the propositions of mathematics, their account of its concepts, for example the number 2, immediately follows. '2' is a meaningless mark occurring in these meaningless formulae. But, whatever may be thought of this as an account of mathematical concepts, it is obviously hopeless as a theory of mathematical concepts; for these occur not only in mathematical propositions, but also in those of everyday life. Thus '2' occurs not merely in '2 + 2 = 4', but also in 'It is 2 miles to the station', which is not a meaningless formula, but a significant proposition, in which '2' cannot conceivably be a meaningless mark. Nor can there be any doubt that '2' is used in the same sense in the two cases, for we can use '2 + 2 = 4' to infer from 'It is two miles to the station and two miles on to the Gogs' that 'It is four miles to the Gogs via the station', so that these ordinary meanings of two and four are clearly involved in '2 + 2 + 4'. So the hopelessly inadequate formalist theory is, to some extent, the result of considering only the propositions of mathematics and neglecting the analysis of its concepts, on which additional light can be thrown by their occurrence outside mathematics in the propositions of everyday life.

[F.P. Ramsey: "The Foundations of Mathematics", Proc. Lond. Math. Soc., Vol.s2-25, Issue 1 (1926) 338-384] http://www.hist-analytic.org/Ramsey.htm http://www-history.mcs.st-and.ac.uk/Mathematicians/Ramsey.html

§ 157 Finitism is usually regarded as the most conservative standpoint for the foundations of mathematics. Induction is justified by appeal to the finitary credo: for every number x there exists a numeral d such that x is d. It is necessary to make this precise. We cannot express it as a formula of arithmetic because "there exists" in "there exists a numeral d" is a metamathematical existence assertion, not an arithmetical formula beginning with $. The finitary credo can be formulated precisely using the concept of the standard model of arithmetic: for every element ξ of Ù there exists a numeral d such that it can be proved that d is equal to the name of ξ, but this brings us into set theory. The finitary credo has an infinitary foundation. {{There is a sober fraction of set theory, namely finite set theory.}} The use of induction goes far beyond the application to numerals. It is used to create new kinds of numbers (exponential, superexponential, and so forth) in the belief that they already exist in a completed infinity. If there were a completed infinity Ù consisting of all numbers, then the axioms of {{PA}} would be valid assertions about numbers and {{PA}} would be consistent. [E. Nelson: "Outline, Against finitism"] http://www.math.princeton.edu/~nelson/papers/outline.pdf

§ 158 One user of transfinite set theory, i.e., a man who arrives where he cannot arrive, is Mohamed El Naschie. http://www.el-naschie.net/bilder/file/Photo-Gallery.pdf

Then came the next quantum jump, around 1990, when M.S. El Naschie who was originally working on elastic and fluid turbulence began to work on his Cantorian version of space- time. He showed that the n-dimensional triadic Cantor set has the same Hausdorff dimension as the dimension of a random inverse golden mean Sierpinski space to the power n-1. [...] Sometime later El Naschie using the work of Prigogine on irreversibility showed that the arrow of time may be explained in a fractal space-time. A few years later two of El Naschie’s papers on the subject were noted by Thompson essential science indicators as the most cited New frontier paper in physics and as Hot paper in engineering. {{That seems to come a bit too early. At the University of Applied Sciences Augsburg the theories of El Naschie have not yet been taught.}} [...] In E-infinity theory El Naschie admit formally infinite dimensional "real” space-time. This infinity is hierarchical in a strict mathematical way and he was able to show that E-infinity has finite number of dimensions when observed from a distance. At low resolution or equivalently at low energy the E-infinity Cantorian space-time appear as a four dimensional space-time manifold. [...] The eigenvalues like equation have a very simple interpretation: Dim E8 E8 = 496 represent all fundamental interactions. Thus it is equal to particle physics 339 symmetries plus the R(4) = 20 of gravity plus ¡0. From that we deduce ¡0 = 496 - 339 - 20 = 137. {{The correct value 137.036 has exorcized some number mysticists and numerologists.}} [...] The author is indebted to the many members of the fractal-Cantorian space-time community {{Cantor's idea of countably many body-atoms and uncountably many ether-atoms gains new impetus. The fatal Space-Time-Community grows, it seems, above all limits.}} [L. Marek-Crnjac: "A short history of fractal-Cantorian space-time", Chaos, Solitons and Fractals 41 (2009) 2697–2705] http://www.msel-naschie.com/pdf/cantorian-history.pdf

§ 159 Hellman, Maddy, and Steel are all impressed by the possibility that set-theoretically substantial mathematics might one day be needed in scientific applications. But of course the mere possibility of future applications provides no support whatever for the indispensability argument. Indeed, one could say of virtually any formal system that future applications are possible. We would only find this possibility noteworthy if we had separate reasons for being interested in the particular system in question. This is really the opposite of an indispensability argument, because ZFC is not gaining credibility from its scientific applications — at present it has none — but rather is seen as a good candidate for future applications because evidently it is already felt to be credible on some other grounds. I must add, however, that given our current understanding of basic physics, the prospect of set-theoretically substantial mathematics ever becoming essential to meaningful scientific applications appears extremely unlikely. This should be obvious to anyone with a basic knowledge of mathematical physics and an understanding of the scope of predicative mathematics. An essential incorporation of impredicative mathematics in basic physics would involve a revolutionary shift in our understanding of physical reality of a magnitude which would dwarf the passage from classical to quantum mechanics (after all, both of these theories are completely predicative). I would rate the likelihood of ZFC turning out to be inconsistent as much higher than the likelihood of it turning out to be essential to basic physics. [Nik Weaver: "Is set theory indispensable?" (2009)] http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.1680v1.pdf

In principle, yes, cp. § 154, for instance. But these faint sounds are always shouted down by There's no con- tra-dic-tion! There's no con- tra-dic-tion! There's no con- tra-dic-tion!

§ 160 {{Another application of set theory?}} One of the remarkable observations made by the Voyager 2 probe was of the extremely fine structure of the Saturn ring system. [...] The Voyager 1 and 2 provided startling images that the rings themselves are composed of thousands of thinner ringlets each of which has a clear boundary separating it from its neighbours. This structure of rings built of finer rings has some of the properties of a Cantor set. The classical Cantor set is constructed by taking a line one unit long, and erasing its central third. This process is repeated on the remaining line segments, until only a banded line of points remains. {{Materialized points are certainly not available in the Saturn ring system.}} [H. Takayasu: "Fractals in the physical sciences", Manchester University Press (1990) p. 36] http://books.google.de/books?id=NRYNAQAAIAAJ&pg=PA180&lpg=PA180&dq=Takayasu:+%2 2Fractals+in+the+physical+sciences%22&source=bl&ots=- _jQrSNVTs&sig=ttuEDGX_6381_T2AdLEBT8HIT20&hl=de&sa=X&ei=Yb6GT- iFDcvUsgao6ITPBg&sqi=2&ved=0CDMQ6AEwAQ#v=onepage&q=Takayasu%3A%20%22Fract als%20in%20the%20physical%20sciences%22&f=false

Mandelbrot conjectures that radial cross-sections of Saturn's rings are fat Cantor sets. For supporting evidence, click each picture for an enlargement in a new window. http://classes.yale.edu/fractals/labs/paperfoldinglab/fatcantorset.html http://www.youtube.com/watch?v=Ztgqa_5vumI

§ 161 {{Yet another application of set theory?}} I propose here, then, first to illustrate, and then to discuss theoretically, the nature and ideal outcome of any recurrent operation of thought, and to develope, in this connection, what one may call the positive nature of the concept of Infinite Multitude.

Prominent among the later authors who have dealt with our problem from the mathematical side, is George Cantor. [...] With this theory of the Mächtigkeiten I shall have no space to deal in this paper, but it is of great importance for forming the conception of the determinate Infinite.

A map of , contained within England, is to represent, down to the minutest detail, every contour and marking, natural or artificial, that occurs upon the surface of England.

Our map of England, contained in a portion of the surface of England, involves, however, a peculiar and infinite development of a special type of diversity within our map. For the map, in order to be complete, according to the rule given, will have to contain, as a part of itself, a representation of its own contour and contents. In order that this representation should be constructed, the representation itself will have to contain once more, as a part of itself, a representation of its own contour and contents; and this representation, in order to be exact, will have once more to contain an image of itself; and so on without limit. We should now, indeed, have to suppose the space occupied by our perfect map to be infinitely divisible, even if not a continuum.

That such an endless variety of maps within maps could not physically be constructed by men, and that ideally such a map, if viewed as a finished construction, would involve us in all the problems about the infinite divisibility of matter and of space, I freely recognize.

Suppose that, for an instance, we had accepted this assertion as true. Suppose that we then attempted to discover the meaning implied in this one assertion. We should at once observe that in this one assertion, "A part of England perfectly maps all England, on a smaller scale," there would be implied the assertion, not now of a process of trying to draw maps, but of the contemporaneous presence, in England, of an infinite number of maps, of the type just described. The whole infinite series, possessing no last member, would be asserted as a fact of existence.

We should, moreover, see how and why the one and the infinitely many are here, at least within thought's realm, conceptually linked. Our map and England, taken as mere physical existences, would indeed belong to that realm of "bare external conjunctions." Yet the one thing not externally given, but internally self-evident, would be that the one plan or purpose in question, namely, the plan fulfilled by the perfect map of England, drawn within the limits of England, and upon a part of its surface, would, if really expressed, involve, in its necessary structure, the series of maps within maps such that no one of the maps was the last in the series.

This way of viewing the case suggests that, as a mere matter of definition, we are not obliged to deal solely with processes of construction as successive, in order to define endless series. A recurrent operation of thought can be characterized as one that, if once finally expressed, would involve, in the region where it had received expression, an infinite variety of serially arranged facts, corresponding to the purpose in question.

[Josiah Royce: "The world and the individual", MacMillan, London (1900) p. 500ff] http://www.archive.org/stream/worldindividual00royciala#page/n0/mode/2up http://www.archive.org/stream/worldindividual00royciala/worldindividual00royciala_djvu.txt

The repeated application of the fotocopier has been proposed as a cheap replacement for expensive electron microscopes. Unfortunately I have forgotten the name of the inventor of this idea.

§ 162 About limits of real sequences. The limit of an infinite sequence (ak) of real numbers ak is determined solely by the finite terms of the sequence. Otherwise, the limit would not have to be computed but would have to be created. Analysis is concerned with analyzing, i.e., with finding. To give an example, we can state with absolute certainty that in the real numbers the sequence 0.1, 0.11, 0.111, ... has the limit 0.111... = 1/9. That is independent of the method which is used to analyze the sequence. But there are different aspects of the limit, namely the numerical value of the limit, the set of coefficients of the power series, its cardinal number, the set of indexes which belong to a digit 1, its cardinal number, the set of indexes which belong to a digit 2, its cardinal number, the set of different digits appearing in the limit, and many further aspects. If any of these aspects is computed by another than the analytical method and turns out as deviating from the analytical result, then the other method is not suitable for analytical purposes.

§ 163 First hidden necessary condition of Cantor's proof. - In the middle of the XX c., meta- mathematics announced Cantor's set theory "naive" and soon the very mention of the term "actual infinity" was banished from all meta-mathematical and set theoretical tractates. The ancient logical, philosophical, and mathematical problem, which during millenniums troubled outstanding minds of humankind, was "solved" according to the principle: "there is no term - there is no problem". So, today we have a situation when Cantor's theorem and its famous diagonal proof are described in every manual of axiomatic set theory, but with no word as to the "actual infinity". However, it is obvious that if the infinite sequence of Cantor's proof is potential then no diagonal method will allow to construct an individual mathematical object, i.e., to complete the infinite binary sequence. Thus, just the actuality of the infinite sequence is a necessary condition (a Trojan Horse) of Cantor's proof, and therefore the traditional, set- theoretical formulation of Cantor's theorem is, from the standpoint of classical mathematics, simply wrong and must be re-written as follows without any contradiction with any logic. [A.A. Zenkin: "Scientific Intuition of Genii Against Mytho-'Logic' of Cantor‘s Transfinite Paradise" Procs. of the International Symposium on “Philosophical Insights into Logic and Mathematics,” Nancy, France, 2002, p. 2] http://www.ccas.ru/alexzen/papers/CANTOR-2003/Zenkin%20PILM2002.doc

§ 164 Because of this obsession with "rigorous" (or "formal") proofs, Mathematics has gotten so specialized, where no one can see the forest, and even most people can't see the whole tree they sit on. All they can see is their tiny branch. Even in specialized conferences, many people skip the invited talks and only go to their own doubly-specialized session. […] Computer algebra, and experimental mathematics, has the potential to become the new unifying "religion". There is still room for some proofs, especially nice ones […] but "formal" proofs should lose their centrality. They are an obsolete relic from a bygone age, just like print- journals, and using a typist to convert your hand-written manuscript to a .tex file. There is so much mathematical knowledge out there that can be discovered empirically (like in the natural sciences, of course it should still be theory-laden, or else it won't go very far). Once we convert to this new religion, we would understand the big picture so much better, and have much more global insight (those that tell me that the purpose of proofs is insight make me laugh, true, the top one percent of proofs give you (local) insight, but the bottom 99 percent are just formal verifications, many of which can already be done by computer, and the rest soon will be [if you are stupid enough to want them]). Proofs are Dead, Long Live Algorithms (and Meta-Algorithms!). [Doron Zeilberger: "Opinion 113: Mathematics is Indeed a Religion, But It has too Many Sects! Let's Unite Under the New God of Experimental Mathematics" (2011)] http://www.math.rutgers.edu/~zeilberg/Opinion113.html

§ 165 Thus the conquest of actual infinity may be considered an expansion of our scientific horizon no less revolutionary than the Copernican system or than the theory of relativity, or even of quantum and nuclear physics. [.A. Fraenkel, A. Levy: "Abstract Set Theory" North Holland, Amsterdam (1976), p. 240]

Inaccessible cardinals on the same scientific level with transistor, laser, computer, NMR, GPS, or space probes? No, rather with Star Trek or Dallas or, as it is a genuinely German invention, with Schwarzwaldklinik.

§ 166 The fact that some discrete items might lack a determinate number, this being connected with the possibility of them being given as a complete whole, was, of course, the traditional, Aristotelian point of view, which Intuitionists, more recently, have still held to. But many others now doubt this fact. Is there any way to show that Aristotle was right? I believe there is.

For when discrete items do clearly collect into a further individual, and we have a finite set, then we determine the number in that set by counting. But what process will determine what the number is, in any other case? The newly revealed independence of the Continuum Hypothesis shows there is no way to determine the number in certain well known infinite sets. [...] The key question therefore is: if there is a determinate number of natural numbers, then by what process is it determined? Replacing 'the number of natural numbers' with 'Aleph zero' does not make its reference any more determinate. The natural numbers can be put into one-one correspondence with the even numbers, it is well known, but does that settle that they have the same number? We have equal reason to say that they have a different number, since there are more of them. So can we settle the determinate number in a set of discrete items just by stipulation?

Indeed, if all infinite sets could be put into one-one correspondence with each other, one would be justified in treating the classification 'infinite' as an undifferentiated refusal of numerability. But given Cantor's discovery that there are infinite sets which cannot be put into one-one correspondence with each other, this conclusion is less compelling.

For Dedekind defined infinite sets as those that could be put into one-one correlation with proper subsets of themselves, so the criteria for 'same number' bifurcate: if any two such infinite sets were numerable, then while, because of the correlation, their numbers would be the same, still, because there are items in the one not in the other, their numbers would be different. Hence such 'sets' are not numerable, and one-one correlation does not equate with equal numerosity [...]

[H. Slater: "The Uniform Solution of the Paradoxes" (2004)] http://www.philosophy.uwa.edu.au/about/staff/hartley_slater/publications/the_uniform_solution_o f_the_paradoxes

§ 167 If, in order to decorate one of my books, https://portal.dnb.de/opac.htm?method=showFirstResultSite¤tResultId=auRef%3D10963 5876%26any&selectedCategory=any I had a straight choice between aleph and the Ishango-bone http://en.wikipedia.org/wiki/Ishango_bone I would choose the latter. Contrary to the former the bone contains mathematics.

§ 168 "However, this negative attitude towards Cantor's set theory, and toward classical mathematics, of which it is a natural generalization, is by no means a necessary outcome of a closer examination of their foundations, but only the result of a certain philosophical conception of the nature of mathematics, which admits mathematical objects only to the extent in which they are interpretable as our own constructions of our own mind, or at least, can be completely given in mathematical intuition. For someone who considers mathematical objects to exist independently of our constructions and of our having an intuition of them individually, and who requires only that the general mathematical concepts must be sufficiently clear for us to be able to recognize their soundness and the truth of the axioms concerning them, there exists, I believe, a satisfactory foundation of Cantor´s set theory in its whole original extent and meaning." [K. Gödel: "Collected Works II" (1964) OUP/1990, p. 258]

{{Nevertheless (cp. § 026) Gödel held Robinson in high esteem and favoured Robinson as his successor.}}: Meanwhile, he was meeting with Gödel at Princeton, where they discussed their mutual interests in mathematics and logic. Gödel was especially impressed by nonstandard analysis and its potential applications in other parts of mathematics. He had suggested in fact that Robinson come to the Institute for an extended period of time, and even hoped that Robinson might one day be his successor. [Robinson to Gödel, April 14, 1971; Gödel papers #011957, Princeton University archives; cited in Dauben, 1995, p. 458]

§ 169 If God has mathematics of his own that needs to be done, let him do it himself. [Errett Bishop: "Foundations of constructive analysis", McGraw-Hill, New York, (1967) Introduction]

Can He have a list of all real numbers? If not, can He store the real numbers in His Infinite Eternity or Eternal Infinity? - each number pinned down by a rational space-time-quadruple? - or quintuple or even centuple? If so, then He could also compute 2 + 2 = 8 and he could conclude that Cantor was right. It does not seem that He has mathematics of His own.

§ 170 The infinite triangle formed by the sequence

0.1 0.11 0.111 ... has height ¡0 but width less than ¡0 (because the limit 1/9, the first line with ¡0 digits, does not belong to the triangle). This lack of symmetry is disturbing for a physicist.

But it would be completely unclear, what side of the infinite triangle is the first one to complete infinity ¡0, when constructed in the following manner:

d d c dc dc a a ac dac dac ... bb bbc dbbc dbbc eeeee

§ 171 Every set can be well-ordered. [E. Zermelo: "Beweis, daß jede Menge wohlgeordnet werden kann", Math. Ann. 59 (1904) 514 Zermelo, E.: "Neuer Beweis für die Möglichkeit einer Wohlordnung", Math. Ann. 65 (1908) 107] Well-ordering elements requires to identify them. For that sake we need physical elements or at least labels, less than 10100 of which are available in the whole universe. But well-orderability is claimed for uncountable sets too. Does enlightenment never touch matheology? Is that degree of energy saving necessary???

§ 172 Wallis in 1684 […] accepts, without any great enthusiasm, the use of Stevin's decimals. He still only considers finite decimal expansions and realises that with these one can approximate numbers (which for him are constructed from positive integers by addition, subtraction, multiplication, division and taking nth roots) as closely as one wishes. However, Wallis understood that there were proportions which did not fall within this definition of number, such as those associated with the area and circumference of a circle:

Real numbers became very much associated with magnitudes. No definition was really thought necessary, and in fact the mathematics was considered the science of magnitudes. Euler, in Complete introduction to algebra (1771) wrote in the introduction: "Mathematics, in general, is the science of quantity; or, the science which investigates the means of measuring quantity." He also defined the notion of quantity as that which can be continuously increased or diminished and thought of length, area, volume, mass, velocity, time, etc. to be different examples of quantity. All could be measured by real numbers.

Cauchy, in Cours d'analyse (1821), did not worry too much about the definition of the real numbers. He does say that a real number is the limit of a sequence of rational numbers but he is assuming here that the real numbers are known. Certainly this is not considered by Cauchy to be a definition of a real number, rather it is simply a statement of what he considers an "obvious" property. He says nothing about the need for the sequence to be what we call today a Cauchy sequence and this is necessary if one is to define convergence of a sequence without assuming the existence of its limit.

[J.J. O'Connor and E.F. Robertson: "The real numbers: Stevin to Hilbert"] http://www-history.mcs.st-and.ac.uk/HistTopics/Real_numbers_2.html

§ 173 The question of language is also particularly relevant, for Brouwer would continue to emphasize that mathematics does not depend on language or logic, being prior to language and logic. Language is merely an instrument of social domination, and it makes impossible a real communication: “nobody has ever communicated his to someone else by means of language.” In 1908, this line of thought would derive into a denounce of classical logic and of axiomatic systems, which obviously from his standpoint cannot be the real foundation of mathematics. [...] Brouwer stressed that he had been elaborating these ideas since 1907, before his involvement with topology, and mentioned how (in his opinion) they also forced him to disagree with Hilbert’s conviction that all mathematical problems are solvable. He emphasized that the foundations for set theory provided both by the logicists and Zermelo were to be rejected. [José Ferreirós: "Paradise Recovered? Some Thoughts on Mengenlehre and ", (2008)]

§ 174 Literature, art, jurisprudence, medicine, and religion are not restricted by reality? But if someone is going to write a novel which on 10100 pages describes 10100 characters, if someone announces a painting in the style of van Gogh with 10100 strokes, if someone expects that the German tax law will contain 10100 paragraphs by the end of the millennium, if someone trusts in the dilution D100 as most helpful in homoeopathy, if someone believes in a final battle between 10100 apes on earth at the end of time --- then he will be considered a fool. But if someone "proves" with mathematical certainty that 10100 different elements can be distinguished and well-ordered, and if this same person is demanding to be taken serious as a scientist --- what can we say?

§ 175 Until then, no one envisioned the possibility that infinities come in different sizes, and moreover, mathematicians had no use for “actual infinity.” The arguments using infinity, including the of Newton and Leibniz, do not require the use of infinite sets. T. Jech: "Set Theory", Stanford Encyclopedia of Philosophy (2002) http://plato.stanford.edu/entries/set-theory/

There are only countably many names. An uncountable set of names cannot be well-ordered - because it does not exist. A set of numbers cannot be well-ordered unless all the numbers have names. This seems to contradict Cantor's diagonal argument - but only if infinite set are complete.

Conclusion: Infinities do not come come in different sizes. In fact mathematicians have never had use for actual infinity because they could not. All they could is to believe that they had use for actual infinity, i. e., for numbers that have no names and cannot be used. --- That's called matheology.

§ 176 Here's a paradox of infinity noticed by Galileo in 1638. It seems that the even numbers are as numerous as the evens and the odds put together. Why? Because they can be put into one-to-one correspondence. The evens and odds put together are called the natural numbers. The first even number and the first natural number can be paired; the second even and the second natural can be paired, and so on. When two finite sets can be put into one-to-one correspondence in this way, they always have the same number of members.

Supporting this conclusion from another direction is our intuition that "infinity is infinity", or that all infinite sets are the same size. If we can speak of infinite sets as having some number of members, then this intuition tells us that all infinite sets have the same number of members.

Galileo's paradox is paradoxical because this intuitive view that the two sets are the same size violates another intuition which is just as strong {{and as justified! If it is possible to put two sets A and B in bijection but also to put A in bijection with a proper subset of B and to put B in bijection with a proper subset of A, then it is insane to judge the first bijection as more valid than the others and to talk about equinumerousity of A and B.}}

[Peter Suber: "Infinite Reflections", St. John's Review, XLIV, 2 (1998) 1-59] http://www.earlham.edu/~peters/writing/infinity.htm#galileo

§ 177 {{I have often denied any benefit of the distinction between countable and uncountable. But the advantage for this particular application cannot be overlooked:}} English nouns are often described as "countable" or "uncountable". http://www.englishclub.com/grammar/nouns-un-countable.htm

§ 178 The whole history of the Mathematische Annalen conflict was quietly incorporated into the oral tradition of European mathematics. Little is known of the aftermath; the Göttinger had won the battle, and they may have been tempted to pick a bone or two with some of the minor actors. […] For Brouwer the matter had, in my opinion, far more serious consequences. His mental state could, under severe stress, easily come dangerously close to instability. Hilbert's attack, the lack of support from old friends, the (real or imagined) shame of his dismissal, the cynical ignoring of his undeniable efforts for the Annalen; each and all of these factors drove Brouwer to a self- chosen isolation. […] After the Annalen affair, little zest for the propagation of intuitionism was left in Brouwer; he continued to work in the field, but on a very limited scale with only a couple of followers. Actually, his whole mathematical activity became rather marginal for a prolonged period. During the thirties Brouwer hardly published at all (only two small papers on topology); he undertook all kinds of projects that had nothing to do with mathematics or its foundations. [Dirk van Dalen: The Mathematical Intelligencer 12,4 (1990) 17-31]

§ 179 The great fascination that contemporary mathematical logic has for its devotees is due, in large measure, to the ever increasing sophistication of its techniques rather than to any definitive contribution to our understanding of the foundations of mathematics. Nevertheless, the achievements of logic in recent years are relevant to foundational questions and it behooves the logician, at least once in a while, to reflect on the basic nature of his subject and perhaps even to report his conclusions. In an address given some years ago the present writer stated a point of view on the foundations of mathematics which may be summed up as follows. (1) Infinite totalities do not exist and any purported reference to them is, literally, meaningless; (2) this should not prevent us from developing mathematics in the classical vein involving the free use of infinitary concepts; and (3) although an infinitary framework such as set theory, or even only Peano number theory cannot be regarded as the ultimate foundation for mathematics, it appears that we have to accept at least a rudimentary form of logic and arithmetic as common to all mathematical reasoning. [A. Robinson: "From a formalist's point of view", Dialectica 23 (1969) 45-49] http://onlinelibrary.wiley.com/doi/10.1111/j.1746-8361.1969.tb01177.x/abstract?

§ 180 A. S. Yessenin-Volpin: "About infinity, finiteness and finitization (in connection with the foundations of mathematics)", Lecture Notes in Mathematics, 873 (1981) pp. 274-313 http://www.springerlink.com/content/76q2110gx555h660/ Probably interesting literature but certainly painstaking to understand. I have not read it and cannot judge whether reading it is rewarding. A much simpler, but as valid conclusion is this: Every form of information transfer (and what else are sequences of digits?) requires an end of file signal. Infinite sequences of digits are therefore unsuitable for mathematical purposes. (That they are unsuitable for all other other purposes is well-known anyway.) Example: The decimal representation of 1/3 = 0.333... is never given by an infinite string of digits but it is always given by a finite word with an end signal, a full stop or period in form of a point - it is absolutely indispensable in correct formal expressions, although sometimes, like in the present case, it is following only after a while - namely here .

§ 181 This paper gives a counterexample to the impossibility, by Gödel's second incompleteness theorem, of proving a formula expressing the consistency of arithmetic in a fragment of arithmetic on the assumption that the latter is consistent. This counterexample gives rise to a new type of metamathematical paradox, to be called the Gödel-Wette paradox, which E. Wette claims to have established since some time ago (see [Wette, 1971]: Wette, Eduard W., On new paradoxes in formalized mathematics, Journal of Symbolic Logic, vol. 36, pp. 376-377. [Wette, 1974]: Wette, Eduard W., Contradiction within Pure Number Theory because of a System Internal ’Consistency’-Deduction, International Logic Review, N. 9, 1974, pp. 51-62. ). Nevertheless, our work is independent of Wette's since we have failed to understand the details of his work {{same happened to Paul Bernays some time before, cp. Kalenderblatt 090804 http://www.hs-augsburg.de/~mueckenh/KB/KB%20001-200.pdf Paul Bernays: "Zum Symposium über die Grundlagen der Mathematik", Dialectica, 25:171-195, 1971. (Translation by: Steve Awodey)] http://www.phil.cmu.edu/projects/bernays/Pdf/bernays28_2003-05-19.pdf }} although we recognize the possibility of the correctness of the latter. Furthermore, the Gödel- Wette paradox is not the only foundational anomaly which the framework of our approach has uncovered but new questions concerning the decision problem, completeness problem, truth definitions and the status of Richard's paradox in arithmetic and set theory (including type theory) have arisen as well. This work will, eventually be unified into a single monograph. [A.S. Yessenin-Volpin, C. Hennix: "Beware of the Gödel-Wette paradox", arXiv (2001)] http://arxiv.org/abs/math/0110094

§ 182 Ultrafinitists don’t believe that really large natural numbers exist. The hard part is getting them to name the first one that doesn’t. [John Baez: "The Inconsistency of Arithmetic", September 30, 2011] The problem is not the size of the number but its information contents. On a pocket calculator, you can multiply 1030 by 1050, but you cannot add or multiply two numbers with more than 10 digits. In real life, you can do superexponentiation, but you cannot use a sequence of more than 10100 digits that lack a finite expansion rule like 0.101010… or Σ1/n2. [WM: Re: The Inconsistency of Arithmetic, September 30, 2011] http://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html#c039531

§ 183 Let's suppose that ZFC is inconsistent. Should anyone here feel shame? I don't see why. [Jesse F. Hughes, 28 Sep 2011] Because there cannot be more infinite paths in the Binary Tree than points where paths get distinct, i.e, nodes where they split. It is a very simple calculation:

| o / \

Every point increases the number of distinct paths by 1. A countable number of points limits the set of all distinct paths to a countable number. Therefore the elements of a set of uncountable paths cannot be distinct. Further the subset of real numbers without a finite definition does not allow to choose a certain element from it. It could not be defined - being tantamount to the matheological statement: it could be defined only in a language that nobody can speak, learn, and understand. That makes Zermelo's axiom of choice obsolete - and his "proof" of well-ordering every set too. Quite a lot of simple mistakes to feel ashamed. http://groups.google.com/group/sci.math/msg/5ccdfbcc58a07f66?dmode=source

§ 184 Cantor coined the word "cardinal number" on January 22, 1886 in a letter to Cardinal Johann Baptist Franzelin - presumably in order to change the Cardinal's mind with respect to transfinite numbers. Cantor hoped, still in vain, to get an affirmative response of the Cardinal about transfinite numbers which, even 12 years after their invention (my goodness, how stubborn Cantor must have been), were completely refused by his contemporary mathematicians.

§ 185 Sir Arthur Eddington (1882 - 1944): There was just one place where (Einstein's) theory did not seem to work properly, and that was - infinity. I think Einstein showed his greatness in the simple and drastic way in which he disposed of difficulties at infinity. He abolished infinity. ... Since there was no longer any infinity, there could be no difficulties at infinity. [Eli Maor: "To Infinity and Beyond. A Cultural History of the Infinite", Birkhäuser, Basel (1987) p. 221]

§ 186 Axiom 1: It is possible to choose every subset of a given set and to choose a first element of this subset, unless the chosen subset is empty.

Axiom 2: It is possible to select a subset of natural numbers with cardinality larger than 10 and sum of elements less than 10.

What is the epistemological difference of these axioms which are equally true?

§ 187 Whatever the choice of language, there will only be a countable infinity of possible texts, since these can be listed in size order, and among texts of the same size, in alphabetical order. {{Here is a simple example: 0 1 00 01 10 11 000 ... }} This has the devastating consequence that there are only a denumerable infinitely of such "accessible" reals, and therefore the set of accessible reals has measure zero. So, in Borel's view, most reals, with probability one, are mathematical fantasies, because there is no way to specify them uniquely. Most reals are inaccessible to us, and will never, ever, be picked out as individuals using any conceivable mathematical tool, because whatever these tools may be they could always be explained in French, and therefore can only "individualize" a countable infinity of reals, a set of reals of measure zero, an infinitesimal subset of the set of all possible {{interesting question: what are possible properties of possible/}} reals. Pick a real at random, and the probability is zero that it's accessible - the probability is zero that it will ever be accessible to us as an individual mathematical object. {{How can we pick? By picking it, a real number would be finitely defined already. That means an undefined real number can never be picked mathematically. And with finger or beak nobody could succeed.}} [Gregory Chaitin: "How real are real numbers?" (2004)] http://arxiv.org/abs/math.HO/0411418

The enumeration of all rational numbers is tantamount to an infinite sum of units. One gets the divergent sequence of all finite cardinal numbers and maintains that a limit exists. That is a mistake. The fact that we can count up to every number does not imply that we can count all numbers. After every finite cardinal number there are infinitely many - but not after all. In a similar way it is impossible to sum all terms of the series Σ1/2n. But contrary to a diverging sequence, the sequence of partial sums of this series deviates from 1 less and less. Therefore 1 can be called the limit.

§ 188 In 1960 the physicist Eugene Wigner published an influential article on "The unreasonable effectiveness of mathematics in the natural sciences". [E. P. Wigner: "The unreasonable effectiveness of mathematics in the natural sciences", Communications on pure and applied mathematics, 13 (1960)] I counter the claim stated in its title with an interpretation of science in which many of the uses of mathematics are shown to be quite reasonable, even rational, although maybe somewhat limited in content and indeed not free from ineffectiveness. The alternative view emphasizes two factors that Wigner largely ignores: the effectiveness of the natural sciences in mathematics, in that much mathematics has been motivated by interpretations in the sciences, and still is; and the central place of theories in both mathematics and the sciences, especially theory-building, in which analogies drawn from other theories play an important role. [Ivor Grattan-Guinness: "Solving Wigner's Mystery: The Reasonable (Though Perhaps Limited) Effectiveness of Mathematics in the Natural Sciences" Springer Science+Business Media, Inc., Volume 30, Number 3 (2008)]

All correct mathematics has to orient itself by means of reality, i.e., natural sciences. Mathematics is applied physics. Cantor intended to follow that scheme with his transfinite set theory, which he, by his own protestation, had devised in order to apply it in natural sciences. Alas his idea of reality was so bad (in contrast to most of his contemporaries he rejected atomism and Darwinism), that it could yield only wrong mathematics.

§ 189 The [...] does not mention totalities at all. Russell held there to be a common cause of, and solution to, all the paradoxes of self-reference. He therefore had to manipulate the Liar paradox into a form where the theory of orders could be applied to it. He did this by parsing the Liar sentence as: there is a proposition that I am affirming and that is false, i.e., $ p(I assert p and p is false). If the quantifier in this proposition has order i, it, itself, is of order i+1, and so does not fall within the scope of the quantifier. This breaks the argument to contradiction. Russell's parsing, by insisting that the self-reference involved be obtained by quantification, strikes one as totally artificial. For a start, the Liar does not have to be asserted to generate a contradiction. But, more fundamentally, the self-reference required may be obtained by ways other than quantification, for example, by a demonstrative: this proposition (or sentence) is false

Notoriously, some 40 years after Tarski's proposal, there is no evidence to show that English is a hierarchy of metalanguages - indeed, there is evidence to show that it is not. Nor is there any reason to suppose that the extensions of words like "true" are context-dependent, in the way that, for example "past" is. [...] It is certainly true that the domains of some quantifiers are contextually determined ("everyone has had lunch"); but, equally, those of others are not ("every natural number is odd or even"), and Parsons gives no reason independent of the paradoxes to suppose that the quantifiers in question are context-dependent. Finally, the claim that different sentences of the same non-indexical type can have different truth values is patently ad hoc.

Tarski obtains the fact that the Liar sentence at level n is true at level n+1, and not at level n, purely by definition: the way the hierarchy is defined, the sentence just is a sentence of level n+1, and not n. But, unless this is pure legerdemain, the question remains as to why things should be defined in this way. Thirdly, and crucially, the parameterisation does not avoid the paradox, merely relocates it. [...] Suppose that this is true in some context/ tokening, then it follows that it is not true in that context/tokening. Hence it is true in no contex/tokening. I.e., it is true in this context/tokening, and so in some context/tokening.

[G. Priest: "Beyond the Limits of Thought", Clarendon Press, Oxford (2006) pp. 143, 153ff]

§ 190 The Binary Tree can be constructed by ¡0 finite paths.

0 / \ 1 2 / \ / \ 3 4 5 6 / \ ... 7 ...

But wait! At each level the number of nodes doubles. We start with the (empty) finite path at level 0 and get 2n+1 - 1 finite paths within the first n levels. The number of all levels of the Binary Tree is called ¡0 although there is no level number ¡0. But mathematics uses only the number of terms of the geometric sequence. That results in 2¡0+1 - 1 = 2¡0 finite paths.

¡0 The bijection of paths that end at the same node proves 2 = ¡0.

This is the same procedure with the terminating binary representations of the rational numbers of the unit interval. Each terminating binary representation q = 0,abc...z is an element out of 2¡0+1 - 1 = 2¡0.

Or remember the proof of divergence of the harmonic series by Nicole d'Oresme. He ¡0+1 ¡0 constructed ¡0 sums (1/2) + (1/3 + 1/4) + (1/5 + ... + 1/8) + ... requiring 2 - 1 = 2 natural ¡0 ¡0 numbers. If there were less than 2 natural numbers (or if 2 was larger than ¡0) the harmonic series could not diverge and mathematics would deliver wrong results.

Beware of the set-theoretic interpretation which tries to contradict these simple facts by ¡0 erroneously asserting ¡0 ∫ 2 .

§ 191 The complete infinite Binary Tree can be constructed by first constructing all ¡0 finite paths and then appending to each path all ¡0 finitely definable tails from 000... to 111... This Binary Tree contains ¡0ÿ¡0 = ¡0 infinite paths. If there were further discernible paths, someone should be able to discern one of them. But since all possible combinations of nodes (including all possible diagonals and anti-diagonals of possible Cantor-lists) that can occur in the mathematical discourse already are present, a human being cannot discern anything additional. Matheologians may claim that God can discern more. But God is not present in mathematics. Mathematicians have no pipeline to God, as Brouwer put it. At least God does never reveal mathematical secrets. Or has any reader ever heard God tell a mathematical secret?

§ 192 We first consider the total amount of energy that one can harvest centrally. [...] one finds 67 Emax = 3.5ÿ10 J, comparable to the total rest-mass energy of baryonic matter within today’s horizon. This total accessible energy puts a limit on the maximum amount of information that can be registered and processed at the origin in the entire future history of the Universe. [...] Dividing the total energy by this value yields a limit on the number of bits that can be processed at the origin for the future of the Universe: Information Processed [...] = 1.35ÿ10120. [..] It is remarkable that the effective future computational capacity for any computer in our Universe is finite, although, given the existence of a global event horizon, it is not surprising. Note that if the equation of state parameter w for dark energy is less than -1, implying that the rate of acceleration of the Universe increases with time, then similar although much more stringent bounds on the future computational capacity of the universe can be derived. In this latter case, distributed computing is more efficient than local computing (by a factor as large as 1010 for w = -1.2, for example), because the Hawking-Bekenstein temperature increases with time, and thus one gains by performing computations earlier in time. [...] On a more concrete level, perhaps, our limit gives a physical constraint on the length of time over which Moore’s Law can continue to operate. In 1965 Gordon Moore speculated that the number of transistors on a chip, and with that the computing power of computers, would double every year. Subsequently this estimate was revised to between 18 months and 2 years, and for the past 40 years this prediction has held true, with computer processing speeds actually exceeding the 18 month prediction. Our estimate for the total information processing capability of any system in our Universe implies an ultimate limit on the processing capability of any system in the future, independent of its physical manifestation and implies that Moore’s Law cannot continue unabated for more than 600 years for any technological civilization. {{Not a breathtakingly large number.}} [Lawrence M. Krauss, Glenn D. Starkman: "Universal Limits on Computation" (2004)] http://arxiv.org/PS_cache/astro-ph/pdf/0404/0404510v2.pdf

Therefore it is not only theoretically wrong that a process can always be completed when every single step can, but it is already practically impossible to perform a step the identification of which requires more than 10130 bits. At least genuine mathematicans would hesitate to accept steps that in principle are impossible - that is reserved for matheologians and lunatics.

§ 193 {{In 1927 David Hilbert gave a talk at Hamburg university, where he explained his opinions about the foundations of mathematics.}} It is a great honour and at the same time a necessity for me to round out and develop my thoughts on the foundations of mathematics, which was expounded here one day five years ago {{compare Kalenderblatt 101212 to 101214 http://www.hs-augsburg.de/~mueckenh/KB/ }} and which since then have constantly kept me most actively occupied. With this new way of providing a foundation for mathematics, which we may appropriately call a proof theory, I pursue a significant goal, for I should like to eliminate once and for all the questions regarding the foundations of mathematics [...] I have already set forth the basic features of this proof theory of mine on different occasions, in Copenhagen [1922], here in Hamburg [1922], in Leipzig [1922], and in Münster [1925]; in the meantime much fault has been found with it, and objections of all kinds have been raised against it, all of which I consider just as unfair as it can be. [...] Poincaré already made various statements that conflict with my views; above all, he denied from the outset the possibility of a consistency proof for the arithmetic axioms, maintaining that the consistency of the method of mathematical induction could never be proved except through the inductive method itself. [...] Regrettably Poincaré, the mathematician who in his generation was the richest in ideas and the most fertile, had a decided prejudice against Cantor's theory, which prevented him from forming a just opinion of Cantor's magnificent conceptions. Under these circumstances Poincaré had to reject my theory, which, incidentally, existed at that time only in its completely inadequate early stages. Because of his authority, Poincaré often exerted a one-sided influence on the younger generation. {{Not to a sufficient degree, unfortunately. --- Then Hilbert discusses the objections by Russell and Whitehead and finally Brouwer. Hilbert concludes:}} I cannot for the most part agree with their tendency; I feel, rather, that they are to a large extent behind the times, as if they came from a period when Cantor's majestic world of ideas had not yet been discovered. {{A world discovered by a man who was behind his times, who did not recognize atoms in the late 19th century, but rejected evolution, who believed in an infinite set of angels and took the basis of his mathematics from the holy bible: "in infinity and beyond".}} [E. Artin et al. (eds.): "D. Hilbert: Die Grundlagen der Mathematik" (1927). Abh. Math. Seminar Univ. Hamburg, vol. 6, Teubner, Leipzig (1928) 65-85. English translation: J. van Heijenoort: "From Frege to Gödel", Harvard Univ. Press, Cambridge, Mass. (1967) 464-479]

§ 194 For many years I have in the hours of leisure granted me, given much study of the Life and Works of Francis Bacon, who in my eyes is one of the greatest geniuses of . By this I have become persuaded, that the opinion so ridiculed by most scholars, of Francis Bacon being the writer of the Shakespearian Dramas, is founded on truth [...] The proofs, I believe I have found, are purely historical, and I propose gradually to publish all the material in question I have at command. [...] Therein Francis Bacon is designated not only as the Creator of the Elisabethean Period, but indeed is addressed as Shakespeare, for (found in the seventeenth distich) denotes clearly in English or <-Shaker>. [Cantor's Preface of the Resurrecti divi Quirini Francisci Baconi edited by Cantor, 1896, acccording to Purkert, Ilgauds: "Georg Cantor 1845 - 1918", Birkhäuser, Basel (1987) p. 85]

§ 195 The weakest of the "platonistic" assumptions introduced by arithmetic is that of the totality of integers. The tertium non datum for integers follows from it; viz.: if P is a predicate of integers, either P is true of each number, or there is at least one exception. By the assumption mentioned, this disjunction is an immediate consequence of the logical principle of the excluded middle; in analysis it is almost continually applied. For example, it is by means of it that one concludes that for two real numbers a and b, given by convergent series, either a = b or a < b or b < a; and likewise: a sequence of positive rational numbers either comes as close as you please to zero or there is a positive rational number less than all the members of the sequence. At first sight, such disjunctions seem trivial, and we must be attentive in order to notice that an assumption slips in. But analysis is not content with this modest variety of platonism; it reflects it to a stronger degree with respect to the following notions: set of numbers, sequence of numbers, and function. It abstracts from the possibility of giving definitions of sets, sequences, and functions. These notions are used in a "quasi combinatorial" sense, by which I mean: in the sense of an analogy of the infinite to the finite. Consider, for example, the different functions which assign to each member of the finite series 1, 2, ..., n a number of the same series. There are nn functions of this sort, and each of them is obtained by n independent determinations. Passing to the infinite case, we imagine functions engendered by an infinity of independent determinations which assign to each integer an integer, and we reason about the totality of these functions. In the same way, one views a set of integers as the result of infinitely many independent acts deciding for each number whether it should be included or excluded. We add to this the idea of the totality of these sets. Sequences of real numbers and sets of real numbers are envisaged in an analogous manner. From this point of view, constructive definitions of specic functions, sequences, and sets are only ways to pick out an object which exists independently of, and prior to, the construction. The axiom of choice is an immediate application of the quasi-combinatorial concepts in question. {{And all that gets lost if there is no God or if he was too dull to create all real numbers (because he knew he could not remember all - neither in his brain nor in a list). The axiom of choice is natural and obviously correct. But since we can choose only what we can name, the axiom of choice supplies one of the strongest contradictions of uncountable sets.}} [Paul Bernays: "On Platonism in Mathematics", (1934)] http://www.phil.cmu.edu/projects/bernays/Pdf/platonism.pdf

§ 196 Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets. {{No there is a difference. All electrons and planets exist, but ideas do not exist unless someone has them. If all sets would exist as complete sets, then obviously they would exist in the platonic shelter. But then this shelter would contain all sets - and its cardinality would be greater than its cardinality. If, however, the shelter would not exist as a complete set, but only as a class or so, why then should any set be complete?}} And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects' perfectly objective properties, so are statements about numbers and sets. Mathematical truths are therefore discovered, not invented. {{This is a proof by naive belief.}} The most important argument for the existence of abstract mathematical objects derives from and goes as follows {{Gottlob Frege: "Foundations of Arithmetic", Blackwell, Oxford, Translation by J.L. Austin (1953)}}. The language of mathematics purports to refer to and quantify over abstract mathematical objects. And a great number of mathematical theorems are true. But a sentence cannot be true unless its sub-expressions succeed in doing what they purport to do. So there exist abstract mathematical objects that these expressions refer to and quantify over. {{This argument is similar to Kant's ontological proof of God (1763). Contrary to Frege Kant noticed his slip during his lifetime (in 1781).}} [Øystein Linnebo: "Platonism in the Philosophy of Mathematics", Stanford Encyclopedia of Philosophy (2009)] http://plato.stanford.edu/entries/platonism-mathematics/

§ 197 "The global unity of mathematics with religion is central in Plato's work, and in his followers' such as Plotinus and Proclus, but also much later in modern times." [Mathematics and the Divine. A Historical Study edited by Teun Koetsier and Luc Bergmans, Amsterdam, Elsevier, 2005, Hardbound, 716 pp., US $250, ISBN-$3: 978-0-444-50328-2, ISBN-IO: 0-444-50328-5 Rewieved by Jean-Michel Kantor in The Mathematical Intelligencer 30, 4 (2008) 70-71] Compare Goedel's proof of God and Cantor's arguing in favour of uncountable numbers and Hilbert's laudatio of Cantor's work. My often cursed noun matheology does not seem to be really far fetched.

§ 198 How can we distinguish between that infinite Binary Tree that contains only all finite initial segments of the infinite paths and that complete infinite Binary Tree that in addition also contains all infinite paths? th Let Lk denote the k level of the Binary Tree. The set of all nodes of the Binary Tree is given ∞ by the union (LL , ,..., L ) of all finite initial segments (L , L , ..., L ) of the sequence of ∪k=0 12 k 1 2 k levels. It contains (as subsets) all finite initial segments of all infinite paths. Does it contain (as subsets) the infinite paths too? How could both Binary Trees be distinguished by levels or by nodes?

Most mathematicians have no answer and know this. They agree that an impossible task is asked for. But some of them offer really exciting ideas. One of them proposed to distinguish between the trees 2<ω and 2§ω. "Not all nodes of the tree 2§ω are finite. Nodes at level ω are not elements of the binary tree 2<ω, but they are elements of the binary tree 2§ω. And yes", he added, "I can state with confidence that nearly all of the experts here support my ideas on this matter." Another one assisted him, addressing me: "You’ve demonstrated copiously over the years in numerous venues that the indistinguishability of 2<ω and 2§ω is an article of faith for you, and that you are either unwilling or unable to learn better. One tree is 2<ω; the other is 2§ω, which has 2ω as its top level, sitting above the levels of 2<ω."

In case you have not yet figured out what is under discussion, here is a simple explanation: Try to distinguish the set of all terminating decimal fractions and the set of all real numbers of the unit interval by digits. When you have understood, here is another task: Try to explain why Cantor's diagonal argument is said to apply to actually infinite decimal representations only. Try to understand, why I claim that everything in Cantor's list happens exclusively within finite initial segments, such that, in effect, Cantor proves the uncountability of the countable set of terminating decimals.

The Binary Tree can be constructed by a sequence such that in every step one node and with it one finite path is added. If nevertheless all infinite paths exist in the Binary Tree after all nodes have been constructed, then it is obvious that infinite paths can creep in without being noticed. If that is proven possible in the tree, then we can also assume that after every line of a Cantor-list has been constructed and checked to not contain the anti-diagonal, nevertheless all real numbers and all possible anti-diagonals can creep into the list in the same way as the infinite paths have crept into the Binary Tree. [Mathematics StackExchange and MathOverflow, Jan. 23, 2013 (meanwhile deleted)]

§ 199 Gödel makes a rather strong comparison between "the question of the objective existence of the objects of mathematical intuition" and the "question of the objective existence of the outer world" which he considers to be "an exact replica."

Gödel's rejection of Russell's "logical fictions" may be seen as a refusal to regard mathematical objects as "insignificant chimeras of the brain."

Gödel's realism, although similar to that of Locke and zz, places emphasis on the fact that the "axioms force themselves upon us as being true." This answers a question, untouched by Locke and Leibniz, why we choose one system, or set of axioms, and not another; that the choice of a mathematical system is not arbitrary.

Gödel, in the "Supplement to the Second Edition" of "What is Cantor's Continuum Problem?" remarked that a physical interpretation could not decide open questions of set theory, i.e. there was (at the time of his writing {{and that did never change}}) no "physical set theory" although there is a physical geometry.

[Harold Ravitch: "On Gödel's Philosophy of Mathematics"] http://www.friesian.com/goedel/ http://www.friesian.com/goedel/chap-2.htm

§ 200 We know that the real numbers of set theory are very different from the real numbers of analysis, at least most of them, because we cannot use them. But it seems, that also the natural numbers of analysis 1, 2, 3, ... are different from the cardinal numbers 1, 2, 3, ...

This is a result of the story of Tristram Shandy, mentioned briefly in § 077 already, who, according to Fraenkel and Levy ["Abstract Set Theory" (1976), p. 30] "writes his autobiography so pedantically that the description of each day takes him a year. If he is mortal he can never terminate; but if he lived forever then no part of his biography would remain unwritten, for to each day of his life a year devoted to that day's description would correspond."

This result is counter-intuitive, but set theory needs the feature of completeness for the enumeration of all rational numbers. If not all could be enumerated, the equality of cardinality of – and Ù could not be proved.

However recently a formal contradiction with the corresponding limit of real analysis could be shown here: http://planetmath.org/?op=getobj&from=objects&id=12607 and here: http://www.hs- augsburg.de/medium/download/oeffentlichkeitsarbeit/publikationen/forschungsbericht_2012.pdf on p. 242 - 244 The limit of remaining undescribed days is infinite according to analysis whereas Fraenkel's story is approved by set theory.

Nevertheless, matheologians violently deny every contradiction. One of them, Michael Greinecker (as a self-proclaimed watchdog and bouncer in MathOverflow http://meta.mathoverflow.net/discussion/1296/crank-post-to-flag-as-spam/#Item_0 an interbreeding of Tomás de Torquemada and Lawrenti Beria) stated: "there is no contradiction. Just a somewhat surprising result. And there is no a apriory reason why one should be able to plug in cardinal numbers in arithmetic formulas for real numbers and get a sensible result."

This means the finite positive integers differ significantly from the finite positive cardinals or the finite positive integers, as Cantor called them. Well, maybe, sometimes evolution yields strange results. But if they differ, how can set theory any longer be considered to be the basis of analysis?

§ 201 Two Commandments of Matheology with Explanations

First Commandment: Ù contains not more than all (finite initial segments {0, 1, 2, ..., n} of the sequence of) natural numbers.

n k Explanation: TRUNC(real) can always be written as ∑a k ⋅10 . k =0

Second Commandment: Ù contains more than all (finite initial segments {0, 1, 2, ..., n} of the sequence of) natural numbers.

n −k Explanation: FRAC(real) can not always be written as ∑a −k ⋅10 . k =1

§ 202 The bulk of Frege's critique of Hilbert consists of criticizing Hilbert's lack of terminological clarity, {{That kind of accusation occurs often. Something is accused to be not formal enough or unclear or completely meaningless. This objection is always advisable if the text is not comprehensible by the reader or inacceptable to him, however without a counter argument being available presently or in general. Hint: I do not want to judge in the Frege-Hilbert controversy. I only aim at the dwarfs who believe to stand on the shoulders of these giants.}} particularly as this applies to the differences between sentences and various collections of thoughts. He takes Hilbert to task for misleadingly using the same sentences to express different thoughts, and points out repeatedly that Hilbert's use of axioms as definitions needs considerably more-careful treatment than Hilbert affords it. The more-substantial criticism flows naturally from this terminological critique: Frege takes it that once one disentangles Hilbert's terminology, it becomes clear that he is simply not talking about the axioms of geometry at all, since the sets of thoughts he actually deals with are the misleadingly-expressed thoughts about e.g. real numbers. And, adds Frege, one cannot infer the consistency of the geometric axioms proper from that of the thoughts Hilbert treats. [Patricia Blanchette (2009)] http://plato.stanford.edu/entries/frege-hilbert/

§ 203 Differences of "all" and "every" in impredicative statements about infinite sets.

Consider the following statements:

A) For every natural number n, P(n) is true. B) There does not exist a natural number n such that P(n) is false. C) For all natural numbers P is true.

A implies B but A does not imply C.

Examples for A: 1) For every n œ Ù, there is m œ Ù with n < m. 2) For every n œ Ù, the set (1, 2, ..., n) is finite. 3) For every n œ Ù, the construction of the first n nodes of a tree adds n paths to the tree. 4) For every n œ Ù, the anti-diagonal of a Cantor-list is not in the lines L1 to Ln.

§ 204 Of today's literature on the foundations of mathematics, the doctrine that Brouwer advanced and called intuitionism forms the greater part. Not because of any inclination for polemics, but in order to express my views clearly and to prevent misleading, conceptions of my own theory, I must look more closely into certain of Brouwer's assertions. Brouwer declares (just as Kronecker did in his day) that existence statements, one and all, are meaningless in themselves unless they also contain the construction of the object asserted to exist; for him they are worthless scrip, and their use causes mathematics to degenerate into a game. [...] What, now, is the real state of affairs with respect to the reproach that mathematics would degenerate into a game? [...] The formula game that Brouwer so depreciates has, besides its mathematical value {{matheology à la Banach-Tarski-paradox contains no value at all - the mathematical value of the proofs that Hilbert believs in can at most be measured in small fractions of lira which not even a single cent will be paid for}}, an important general philosophical significance. For this formula game is carried out according to certain definite rules, in which the technique of our thinking is expressed. These rules form a closed system that can be discovered and definitively stated {{like other religious systems too: Buddhism, Christianity, Hinduism, Islam, Judaism, ... In Islam they enjoy apostasy-punishment by death. Heretics are usually killed. In matheology heretics are called cranks and attempts are made to remove them from their academic poitions. So matheology is not quite as intolerant as Islam but less tolerant than Buddhism and Hinduism. There is another parallel: Moslems are not allowed to contact God in any other language than the Arabic. Allah seems to be less educated than Jahwe or God who accept prayers in every language. In matheology every important prayer must be uttered in a certain . The God of matheology and his disciples seem to be very limited too.}} [E. Artin et al. (ed.): "D. Hilbert: Die Grundlagen der Mathematik" (1927). Abh. Math. Seminar Univ. Hamburg, Bd. 6, Teubner, Leipzig (1928) 65-85. English translation in J. van Heijenoort: "From Frege to Gödel", Harvard Univ. Press, Cambridge, Mass. (1967) 464-479.] http://www.marxists.org/reference/subject/philosophy/works/ge/hilbert.htm

§ 205 The fundamental idea of my proof theory is none other than to describe the activity of our understanding, to make a protocol of the rules according to which our thinking actually proceeds. Thinking, it so happens, parallels speaking and writing: we form statements and place them one behind another. If any totality of observations and phenomena deserves to be made the object of a serious and thorough investigation, it is this one - since, after all, it is part of the task of science to liberate us from arbitrariness, sentiment, and habit and to protect us from the that already made itself felt in Kronecker's views and, it seems to me, finds its culmination in intuitionism. {{The erroneous opinion that a sphere never doubles itself really should be opposed with severeness! Folks, distribute marbles or blow up balloons and condoms in order to prove that!}} Intuitionism's sharpest and most passionate challenge is the one it flings at the validity of the principle of excluded middle [...] Existence proofs carried out with the help of the principle of excluded middle usually are especially attractive because of their surprising brevity and elegance. Taking the principle of excluded middle from the mathematician would be the same, proscribing the telescope to the astronomer or to the boxer the use of his fists. [...] Not even the sketch of my proof of Cantor's continuum hypothesis has remained uncriticised. I would therefore like to make some comments on this proof. [E. Artin et al. (ed.): "D. Hilbert: Die Grundlagen der Mathematik" (1927). Abh. Math. Seminar Univ. Hamburg, Bd. 6, Teubner, Leipzig (1928) 65-85. English translation in J. van Heijenoort: "From Frege to Gödel", Harvard Univ. Press, Cambridge, Mass. (1967) 464-479.]

{{The interests of marxists are rather philosophical and social and less mathematical. But with respects to Hilbert's "proof of the continuum hypothesis" they have hit the nail right on the head:}} "The whole of Hilbert selection for series reproduced here, minus some inessential mathematical formalism." http://www.marxists.org/reference/subject/philosophy/works/ge/hilbert.htm (For an evaluation of Hilbert's logic by Zermelo compare § 117.)

§ 206 From my presentation you will recognise that it is the consistency proof that determines the effective scope of my proof theory and in general constitutes its core. The method of W. Ackermann permits a further extension still. For the foundations of ordinary analysis his approach has been developed so far that only the task of carrying out a purely mathematical proof of finiteness remains. Already at this time I should like to assert what the final outcome will be: mathematics is a presuppositionless science. To found it I do not need God {{I do not need Hilbert, said Gott - and created Gödel.}}, as does Kronecker {{There is some correction required: Kronecker's sentence "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk" probably has been meant ironically like the sentence "Gott Würfelt nicht" of the pronounced atheist Albert Einstein. Hilbert, on the other hand needs God, among others for his paradise.}}, or the assumption of a special faculty of our understanding attuned to the principle of mathematical induction, as does Poincaré {{Here we need nothing but a little bit of common sense: If a theorem is valid for the number k, and if from its validity for the number n + k the validity for n + k + 1 can be concluded with no doubt, then n can be replaced by n + 1, and the validity for n + k + 2 is proven too. This is the foundation of mathematics. To prove anything about this principle is as useless as the proof that 1 + 1 = 2.}}, or the primal intuition of Brouwer, or, finally, as do Russell and Whitehead, axioms of infinity, reducibility, or completeness, which in fact are actual, contentual assumptions that cannot be compensated for by consistency proofs. {{That is correct. The axiom of infinity is simply an inconsistent assumption like the assumption of a set of natural numbers with cardinality 10 and sum 10. Therefore it cannot be proved from mathematics. But everything is proved after assuming it as an axiom.}} I would like to note further that P. Bernays has again been my faithful collaborator. He has not only constantly aided me by giving advice but also contributed ideas of his own and new points of view, so that I would like to call this our common work. {{His newest point of view is this: "If we pursue the thought that each real number is defined by an arithmetical law, the idea of the totality of real numbers is no longer indispensable." (Paul Bernays, 1934, cp. § 144)}} [E. Artin et al. (ed.): "D. Hilbert: Die Grundlagen der Mathematik" (1927). Abh. Math. Seminar [E. Artin et al. (ed.): "D. Hilbert: Die Grundlagen der Mathematik" (1927). Abh. Math. Seminar Univ. Hamburg, Bd. 6, Teubner, Leipzig (1928) 65-85. English translation in J. van Heijenoort: "From Frege to Gödel", Harvard Univ. Press, Cambridge, Mass. (1967) 464-479.] http://www.marxists.org/reference/subject/philosophy/works/ge/hilbert.htm

§ 207 Towards the end of his Address on the Unity of Knowledge, delivered at the 1954 bicentennial celebrations, Weyl enumerates what he considers to be the essential constituents of knowledge. At the top of his list comes "…intuition, mind's ordinary act of seeing what is given to it." (Weyl 1954, 629) In particular Weyl held to the view that intuition, or insight - rather than proof - furnishes the ultimate foundation of mathematical knowledge. {{What else should furnish it? A can be given for every stupidity, and be it infinite.}} Thus in his Das Kontinuum of 1918 he says: "In the Preface to Dedekind (1888) we read that 'In science, whatever is provable must not be believed without proof.' This remark is certainly characteristic of the way most mathematicians think. Nevertheless, it is a preposterous principle. As if such an indirect concatenation of grounds, call it a proof though we may, can awaken any 'belief' apart from assuring ourselves through immediate insight that each individual step is correct. In all cases, this process of confirmation - and not the proof - remains the ultimate source from which knowledge derives its authority; it is the 'experience of truth'” (Weyl 1987, 119) {{like Zermelos "proof" of the well- ordering assertion is the experience of untruth}}. [John L. Bell: "Hermann Weyl", Stanford Encyclopedia of Philosophy (2009)] http://plato.stanford.edu/entries/weyl/index.html

§ 208 In Consistency in Mathematics (1929), Weyl characterized the mathematical method as "the a priori construction of the possible in opposition to the a posteriori description of what is actually given". {{Above all, mathematics has to be consistent. And there is only one criterion for consistency: The "model" of reality.}} The problem of identifying the limits on constructing “the possible” in this sense occupied Weyl a great deal. He was particularly concerned with the concept of the mathematical infinite, which he believed to elude “construction” in the naive set-theoretical sense. Again to quote a passage from Das Kontinuum: "No one can describe an infinite set other than by indicating properties characteristic of the elements of the set…. The notion that a set is a 'gathering' brought together by infinitely many individual arbitrary acts of selection, assembled and then surveyed as a whole by consciousness, is nonsensical; 'inexhaustibility' is essential to the infinite." [...] Small wonder, then, that Hilbert was upset when Weyl joined the Brouwerian camp. [John L. Bell: "Hermann Weyl", Stanford Encyclopedia of Philosophy (2009)] http://plato.stanford.edu/entries/weyl/index.html

§ 209 In Das Kontinuum Weyl says: "The states of affairs with which mathematics deals are, apart from the very simplest ones, so complicated that it is practically impossible to bring them into full givenness in consciousness and in this way to grasp them completely." Nevertheless, Weyl felt that this fact, inescapable as it might be, could not justify extending the bounds of mathematics to embrace notions, such as the actual infinite, which cannot be given fully in intuition even in principle. He held, rather, that such extensions of mathematics into the transcendent are warranted only by the fact that mathematics plays an indispensable role in the physical sciences, in which intuitive evidence is necessarily transcended. As he says in The Open World: "… if mathematics is taken by itself, one should restrict oneself with Brouwer to the intuitively cognizable truths … nothing compels us to go farther. But in the natural sciences we are in contact with a sphere which is impervious to intuitive evidence; here cognition necessarily becomes symbolical construction. Hence we need no longer demand that when mathematics is taken into the process of theoretical construction in physics it should be possible to set apart the mathematical element as a special domain in which all judgments are intuitively certain; from this higher standpoint which makes the whole of science appear as one unit, I consider Hilbert to be right" {{me too.}} Weyl soon grasped the significance of Hilbert's program, and came to acknowledge its "immense significance and scope". Whether that program could be successfully carried out was, of course, still an open question. But independently of this issue Weyl was concerned about what he saw as the loss of content resulting from Hilbert's thoroughgoing formalization of mathematics. "Without doubt", Weyl warns, "if mathematics is to remain a serious cultural concern {{a mathematician should never give up this premise}}, then some sense must be attached to Hilbert's game of formulae." [John L. Bell: "Hermann Weyl", Stanford Encyclopedia of Philosophy (2009)] http://plato.stanford.edu/entries/weyl/index.html

§ 210 An accessible number, to Borel, is a number which can be described as a mathematical object. The problem is that we can only use some finite process to describe a real number so only such numbers are accessible. We can describe rationals easily enough, for example either as, say, one-seventh or by specifying the repeating decimal expansion 142857. Hence rationals are accessible. We can specify Liouville's easily enough as having a 1 in place n! and 0 elsewhere. Provided we have some finite way of specifying the n-th term in a Cauchy sequence of rationals we have a finite description of the resulting real number. However, as Borel pointed out, there are a countable number of such descriptions. Hence, as Chaitin writes: "Pick a real at random, and the probability is zero that it's accessible - the probability is zero that it will ever be accessible to us as an individual mathematical object." [J.J. O'Connor and E.F. Robertson: "The real numbers: Attempts to understand"] http://www-history.mcs.st-and.ac.uk/HistTopics/Real_numbers_3.html But how to pick this dark matter of numbers? Only accessible numbers can get picked. Unpickable numbers cannot appear anywhere, neither in mathematics nor in Cantor's lists. Therefore Cantor "proves" that the pickable numbers, for instance numbers that can appear as an antidiagonal of a defined list, i.e., the countable numbers, are uncountable.

§ 211 The belief in the universal validity of the principle of the excluded third in mathematics is considered by the intuitionists as a phenomenon of the history of civilization of the same kind as the former belief in the rationality of π, or in the rotation of the firmament about the earth {{or the assumption that every particle has definite position and velocity at every time.}} The intuitionist tries to explain the long duration of the reign of this dogma by two facts: firstly that within an arbitrarily given domain of mathematical entities the non-contradictority of the principle for a single assertion is easily recognized; secondly that in studying an extensive group of simple every-day phenomena of the exterior world, careful application of the whole of classical logic was never found to lead to error. [This means de facto that common objects and mechanisms subjected to familiar manipulations behave as if the system of states they can assume formed part of a finite discrete set, whose elements are connected by a finite number of relations.] {{Unfortunately this principle, without any justification, has been applied to infinite sets.}} [L.E.J. Brouwer: "Lectures on Intuitionism - Historical Introduction and Fundamental Notions" (1951), Cambridge University Press (1981)] http://www.marxists.org/reference/subject/philosophy/works/ne/brouwer.htm

§ 212 A synopsis of Brouwer's position yields two statements: - Classical mathematics is contradictory. - Infinite remains potential. It just means that you can go on and on. Classical mathematics, i.e., Cantor's set theory cannot be reconciled with constructivism. There are two big lies of matheology. The first is to call transfinite set theory "classical mathematics". The second is to imply that this classical mathematics is as valid as constructivism. This is not a matter of taste but provably wrong as becomes clear from the following statements: According to Hilbert, not to believe in tertium non datur is the most glaring disbelief that we find in the human history. [D. Hilbert: "Die Grundlegung der elementaren Zahlenlehre", Vortrag in Hamburg (1930), Mathematische Annalen 104 (1933) 485-494] Brouwer calls the belief in tertium non datur a disappearing superstition. [Brouwer-lecture in Berlin, reported by A. Weil in Dirk van Dalen: "Mystic, Geometer, and Intuitionist: The Life of L.E.J. Brouwer, Vol. 2", Clarendon Press, Oxford (2005) 643]

§ 213 Zermelo’s proof had not indicated how to determine the covering γ uniquely, and yet one needed to be certain that γ remained the same throughout the proof. How could one be sure? Moreover, even if such a covering γ existed and could be defined, it was doubtful that one could use γ in the way that Zermelo had; for the subsets M’ of M were not defined in a unique way. Indeed, Lebesgue doubted that one would ever be able to state a general method for well- ordering a given set. {{That was very wise. But it shows one fact above all: The possibility of a well-ordering of the reals had been expected within reach at that time. Today its impossibility for the reals is well known. It is not admitted that this is contradicting Zermelo's proof because there cannot be a contradiction in ZFC which stands for Zero Falsifying Contradictions.}} [Gregory H. Moore: "The Origins of Zermelos Axiomatization of Set Theory", Journal of Philosophical Logig 7 (1978) 307-329] http://www.jstor.org/discover/10.2307/30226178?uid=3737864&uid=2&uid=4&sid=21101627338 593

§ 214 What’s wrong with the axiom of choice?

Part of our aversion to using the axiom of choice stems from our view that it is probably not ‘true’. {{In fact it is true for existing sets - but there it is not required as an axiom but is a self- evident truth.}} A theorem of Cohen shows that the axiom of choice is independent of the other axioms of ZF, which means that neither it nor its negation can be proved from the other axioms, providing that these axioms are consistent. Thus as far as the rest of the standard axioms are concerned, there is no way to decide whether the axiom of choice is true or false. This leads us to think that we had better reject the axiom of choice on account of Murphy’s Law that "if anything can go wrong, it will". This is really no more than a personal hunch about the world of sets. We simply don’t believe that there is a function that assigns to each non-empty set of real numbers one of its elements. While you can describe a selection function that will work for finite sets, closed sets, open sets, analytic sets, and so on, Cohen’s result implies that there is no hope of describing a definite choice function that will work for "all" non-empty sets of real numbers, at least as long as you remain within the world of standard Zermelo-Fraenkel set theory. And if you can’t describe such a function, or even prove that it exists without using some relative of the axiom of choice, what makes you so sure there is such a thing? Not that we believe there really are any such things as infinite sets, or that the Zermelo- Fraenkel axioms for set theory are necessarily even consistent. Indeed, we’re somewhat doubtful whether large natural numbers (like 805000, or even 2200) exist in any very real sense, and we’re secretly hoping that Nelson will succeed in his program for proving that the usual axioms of arithmetic -and hence also of set theory - are inconsistent. (See E. Nelson. Predicative Arithmetic. Princeton University Press, Princeton, 1986.) All the more reason, then, for us to stick with methods which, because of their concrete, combinatorial nature, are likely to survive the possible collapse of set theory as we know it today. [Peter G. Doyle, John Horton Conway: "Division by Three" 1994, ARXIV math/0605779v1] http://arxiv.org/abs/math/0605779v1

§ 215 The set of all sets and the devil have one property in common: Both terrify only their believers.

§ 216 One remark that Penelope Maddy makes several times in in Mathematics, is that if the indispensability argument was really important in justifying mathematics, then set theorists should be looking to debates over quantum gravity to settle questions of new axioms. Since this doesn’t seem to be happening, she infers that the indispensability argument can’t play the role Quine and Putnam (and perhaps her earlier book?) argued that it does. [...] I don’t know much about the details, but from what I understand, physicists have conjectured some deep and interesting connections between seemingly disparate areas of mathematics, in order to explain (or predict?) particular physical phenomena. These connections have rarely been rigorously proved, but they have stimulated mathematical research both in pursuing the analogies and attempting to prove them. Although the mathematicians often find the physicists’ work frustratingly imprecise and non-rigorous, once the analogies and connections have been suggested by physicists, mathematicians get very interested as well. If hypothetically, one of these connections was to turn out to be independent of ZFC, I could imagine that there would at least be a certain camp among mathematicians that would take this as evidence for whatever large cardinal (or other) principle was needed to prove the connection. [Kenny Easwaran: "Set Theory and String Theory" (2006)] http://antimeta.wordpress.com/2006/10/29/set-theory-and-string-theory/ Of course, every physical result is independent of ZFC.

§ 217 Whatever may play a role in mathematics, symbols, numbers, operators, definitions, theorems: All together belongs to a countable set. The remaining is matheology. Alas, contrary to theology matheology is not suitable to awaken the hope of happiness and luck after death in - not even in the souls of matheologians.

§ 218 What is Mathematics? Most mathematicians don't know and don't care. Mathematics is what mathematicians do. [...] In fifty years (at most) human mathematicians will be like lamp- lighters and ice-delivery men. All serious math will be done by computers. Let's hope that human philosophy will still survive, but we need to adjust naturalism to the practice of math in the future and to the way it will be done by machines. Of course, we don't know exactly how, so let's put this project of Naturalist mathematical philosophy on hold and wait to see how things turn out in fifty years.

Tim Gowers said that we are all formalists, but most of us don't know it (and if we knew, we wouldn't care). I kind of agree, but this is only a corollary of a more profound truth: Everything is Combinatorics. Classify Lie Algebras? It is just root systems and Dynkin Diagrams. Finite Groups? The Monster is a Combinatorial Design. Even when it is not obviously combinatorics, it could be made so. If it is too hard for us, then we need a computer! But computer science is all Discrete Math, alias combinatorics. In a way Logic is too. But Logic is so low-level, like machine language. It is much more fun and gratifying to work in Maple, and do higher-level combinatorics.

I am also a trivialist. [...] We humans, and even our computers, can only prove trivial results. Since all knowable math is ipso facto trivial, why bother? So only do fun problems, that you really enjoy doing. It would be a shame to waste our short lives doing "important" math, since whatever you can do, would be done, very soon (if not already) faster and better (and more elegantly!) by computers. So we may just as well enjoy our humble trivial work.

[Doron Zeilberger: "Opinion 69: Roll Over Platonism, Logicisim, Formalism, Intuitionism, Constructivism, Naturalism and ! Here Comes Combinatorialism and Trivialism." (2005)]

§ 219 Applied Math is an indispensable part of various engineering disciplines because its application and usefulness in predictive models has been validated against real-world conditions again and again and again. If, for argument’s sake, PA was proven inconsistent, then math merely becomes a defacto natural science like biology or chemistry, in the sense that the “validity” of math no longer stems from axioms, but rather validation against real world conditions and observations. [Paul AC Chang, Re: The Inconsistency of Arithmetic, The n-Category Café, October 2, 2011] http://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html#c039531

§ 220 PA {{Peano-Arithmetik}} already tells us that the universe is infinite, but PA “stops” after we have all the natural numbers. {{No, PA never stops because it never reaches an end. Here potential and actual infinity are confused.}} ZFC goes beyond the natural numbers; in ZFC we can distinguish different infinite cardinalities such as “countable” and “uncountable”, and we can show that there are infinitely many cardinalities, uncountably many, etc. {{and we can show that there is nothing of that kind other than in dreams, but not in logic.}} [Saharon Shelah: "Logical Dreams" (2002)] http://arxiv.org/PS_cache/math/pdf/0211/0211398v1.pdf

§ 221 As I get older I seem to be getting more and more relaxed about foundational issues. I’m happy to see people formalize and explore all imaginable attitudes toward the foundations of mathematics. I feel confident that the more interesting axiom systems will eventually attract more researchers, while the less interesting ones will remain marginal. I am not eager for one system to prevail over all others … nor do I feel any desire for systems I dislike to go completely extinct. It’s a lot like my fondness for biodiversity. I enjoy the diversity of life, and am very happy there are tigers, and would be sad for them to go extinct, even though I wouldn’t want a bunch running around in my back yard. In particular, I’m glad there are ultrafinitists, because I suspect that only someone with views like that could be motivated to prove the inconsistency of (say) Peano arithmetic, and seek plausible strategies for doing it. If everyone believes Peano arithmetic is consistent, and it’s not, we’re in big trouble because it’ll take us a long time to discover it. So we need a few lonely people working on the other side of this issue. I don’t think they’ll succeed, but I’m glad they’re trying. Even if they don’t succeed, there could be some interesting concepts and theorems that only they are likely to find. Finally, I don’t expect these people to take the same ‘relaxed, balanced’ attitude that I have. I suspect that only someone with strong opinions could possibly be motivated to spend a lot of time developing ultrafinitism, or trying to prove the inconsistency of Peano arithmetic. Expecting them to share my relaxed attitude is a bit like expecting a tiger to be an environmentalist. [John Baez: "The Inconsistency of Arithmetic", n-Category-Cafe, September 30, 2011]

§ 222 Consider a Cantor-list with entries an and anti-diagonal d:

" n (for every n) œ Ù: (an1, an2, ..., ann) ∫ (d1, d2, ..., dn). " n (for every n) œ Ù: (an1, an2, ..., ann) is terminating. " n (for every n) œ Ù: (d1, d2, ..., dn) is terminating.

" n (for all n) œ Ù: (an1, an2, ..., ann) ∫ (d1, d2, ..., dn). " n (for all n) œ Ù: (an1, an2, ..., ann) is terminating. " n (for all n) œ Ù: (d1, d2, ..., dn) is not terminating.

That's the origin of matheology.

§ 223 How obvious a contradiction has to result from an additional axiom in order to reject it? The Axiom of Choice (AC) states that every set can be well-ordered. In order to well-order an uncountable set, an uncountable alphabet is required, since a countable alphabet is not sufficient to label uncountably many elements (compare the Binary Tree, § 190). But an alphabet is a linearly ordered set (otherwise you would never find most letters of the alphabet - compare the telephone book). And linear ordering implies well-ordering. So the Axiom of Choice contradicts the other ZF-axioms. (This has already been shown by Hausdorff-Banach-Tarski with the result that by means of AC we can prove that, after some turning and twisting, but without any addition or subtraction of even one single point, the measurable set V is identical with the measurable set 2V.) With equal right we can introduce the Axiom of Meagre Sum (AMS) stating: There is a set of n positive natural numbers with sum nÿn/2. This axiom is not constructive, since nobody can construct such a set. But the disproof by the well known fact that the sum of n different positive natural numbers is never less than n(n+1)/2 is not less obvious than the disproof of AC.

§ 224 Consider a tabletop supported by three legs. The tabletop is stable if its center of mass lies in the triangle formed by the supporting points. No special one of the three points is necessary. But we know, that three legs exist, if the table is stable. If we have support by four (or more) suitable legs, we can show that one (ore more) are superfluous, i.e., not necessary. Could we prove that of three legs one is not required, then reality would be contradictory. But since reality proves its consistency by simple existence (and not by inconsistent sets of axioms), such a proof is impossible.

Now consider the list of finite initial segments of natural numbers 1 1, 2 1, 2, 3 ... According to set theory it contains all ¡0 natural numbers in its lines. But is does not contain a line containing all natural numbers. Therefore it must be claimed that more than one line is necessary to contain all natural numbers. This means at least two lines are necessary. There are no special lines necessary, but there must be at least two. In this case, however, we can prove, by the construction of the list, that every union of a pair of lines is contained in one of the lines. This contradicts the assertion that all natural numbers exist and are in lines of the list.

The solution of this paradox is a potentially infinite list that has always a last line (which is the only line necessary to contain all natural numbers that are in the list) but this last line cannot be fixed.

§ 225 Axiom der Auswahl. - Man kann das Axiom auch so ausdrücken, daß man sagt, es sei immer möglich, aus jedem Elemente M, N, R , ..., von T ein einzelnes Element m, n, r, ... auszuwählen und alle diese Elemente zu einer Menge S1 zu vereinigen. [E. Zermelo: "Untersuchungen über die Grundlagen der Mengenlehre I", Mathematische Annalen 65 (1908) 261-281]

So the axiom of choice says that it is always possible to choose an element from every non- empty set and to union the chosen elements into a set S1. "Choosing" something means pointing to or showing this something, or, if this something has no material existence, defining or labelling it by a finite word. For uncountable sets this is known to be as impossible as to find a second triple besides 3, 5, 7. Would matheologians accept the axiom "there is a second prime number triple" if necessarily required to save matheology? Yes, I am sure.

§ 226 There is an isomorphism between the field of all non-negative binary representations r of real numbers (—, +, ÿ) and the filed of all paths p of the extended Binary Tree (BT, +, ÿ) such that for every r in — there is a p = f(r) in BT (and vice versa) and we have for all a, b, r, s in — (a and b can also be taken from BT): f(ar + bs) = af(r) + bf(s).

The extended Binary Tree is obtained by extending the ordinary Binary Tree to all integers: ... 0 1 0 1 \/ \/ 0 1 \ / . / \ 0 1 /\ /\ 0 1 0 1 ...

§ 227 {{Fields medalist Voevodsky}} stated the theorem as follows [...]: It is impossible to prove the consistency of any formal reasoning system which is at least as strong as the standard axiomatization of elementary number theory ("first order arithmetic"). So he failed to inform his audience that the impossibility that Goedel actually established was the impossibility of proof-in-S of a sentence expressing the consistency of S, for any consistent and sufficiently strong system S. As we know, Gentzen's proofs of the consistency of PA are among the most important results in proof-theory, second only to Goedel's results themselves and perhaps Prawitz' normalization results. (For a great overview of Gentzen's proofs, see von Plato's SEP entry.) What I find most astonishing about Gentzen's proofs, based on transfinite induction, is that the theory obtained by adding quantifier free transfinite induction to primitive recursive arithmetic is not stronger than PA, and yet it can prove the consistency of PA (it is not weaker either, obviously; they just prove different things altogether). One may raise eyebrows concerning transfinite induction (and apparently this is what lies behind Voevodsky's dismissal of Gentzen's results), but apparently most mathematicians and logicians seem quite convinced of the cogency of the proof. {{Most astrologers are convinced of the consistency of astrology.}} [...] So Voevodsky seems to seriously entertain the possibility of PA's inconsistency. Is it because he doesn't understand Goedel's results, or Gentzen's results, or both? Or is there something else going on? [...] Now, within the bigger picture of things, the consistency of PA is actually a tangential, secondary issue. Voevodsky’s seemingly polemic statement concerning the potential inconsistency of PA in fact seems to amount to the following: all the currently available proofs of the consistency of PA in fact rely on the very claim they prove, namely the consistency of PA, on the meta-level. [...] So if PA was inconsistent, these proofs would still go through; in other words, there is a sense in which such proofs are circular in that they presuppose the very fact that they seek to prove. {{And nobody has noticed hitherto?}} [Catarina Dutilh Novaes (2011)] http://m-phi.blogspot.com/2011/05/voevodsky-consistency-of-pa-is-open.html http://m-phi.blogspot.com/2011/06/latest-news-on-inconsistency-of-pa.html

§ 228 From 1969 until 1973 I worked to delineate mathematical methods devoid of any unprovable aspects. I began by observing that some logicians disagree with most mathematicians on one important point: The logicians insist that false assumptions must lead to both the proof and disproof of every meaningful statement within the logical system in question. The math people believe that they can make up sets of axioms that make sense to them (true in the world or not) and that the resultant math will be free of logical flaws. Here are some of the conclusions I came to during this time: 1. Calculus does not need any concept of infinity in order to provide limits, derivatives, integrals, and differentials. 2. No infinity is possible unless axioms assert it. That is, no infinity can be derived without being presupposed. 3. No even roots of -1 are needed except one (i = the square root). 4. All of the above facts are known to many mathematicians. 5. Real numbers that are not rational numbers can be expressed as limits of functions of rational numbers. This means that quantities like π and e need not be regarded as numbers and they may be formally handled just as computer programs handle them. It also means that the concept of a process replaces the real numbers that are not rational. The complex numbers (a + bi) become processes where a and b numbers or processes. 6. Mathematics needs to be synthetic (founded upon definitions which, in turn, are founded upon undefined but perceived meanings in the common language). Axioms are unnecessary and harmful. No axiomatic system works anyway if we don't agree on the meaning of such fundamental terms as single, pair, the, associated with, and the like. Good systems would rely on a minimum of such terms and would explicitly recognize them. Formal math comes from our of reality, not the other way around. 7. The phenomenon of the conditional branch (if incorporated into math proper), represents a giant advance in the power of math to solve problems. That is the short story. We get every kind of functionality in the whole world without any logical flaws and without esoteric and spooky contradictions (many mathematicians are in awe of them, but they can all be easily fixed). [Jim Trek (1999)] http://members.chello.nl/~n.benschop/finite.htm

§ 229 The difference between potential and actual infinity can even be photographed: Infinity, to find use in set theory, must split off. The following pictures of a movie of an everexpanding square show this for the first time:

o

oo oo

ooo ooo ooo

oooo oooo oooo oooo

...

For each finite square we find height = width. For the infinite square however, height > width, namely an infinite sequence of finite lines (scale changed):

......

There seem to be some gravity effects involved in transfinite set theory. Otherwise the (actually) infinite set of natural numbers cannot be gathered.

[WM: "Gravity effects detected in transfinite set theory", sci.logic, sci.math, 8 Oct 2011] http://groups.google.com/group/sci.math/msg/d48b3d3e7581802d?dmode=source

§ 230 Cantor's theory of infinite sets, developed in the late 1800's, was a decisive advance for mathematics, but it provoked raging controversies and abounded in paradox. One of the first books by the distinguished French mathematician Emile Borel (1871-1956) was his Lecons sur la Théorie des Fonctions [Borel, 1950], originally published in 1898, and subtitled Principes de la théorie des ensembles en vue des applications à la théorie des fonctions. This was one of the first books promoting Cantor's theory of sets (ensembles), but Borel had serious reservations about certain aspects of Cantor's theory, which Borel kept adding to later editions of his book as new appendices. The final version of Borel's book, which was published by Gauthier-Villars in 1950, has been kept in print by Gabay. That's the one that I have, and this book is a treasure trove of interesting mathematical, philosophical and historical material. One of Cantor's crucial ideas is the distinction between the denumerable or countable infinite sets, such as the positive integers or the rational numbers, and the much larger nondenumerable or uncountable infinite sets, such as the real numbers or the points in the plane or in space. Borel had constructivist leanings, and as we shall see he felt comfortable with denumerable sets, but very uncomfortable with nondenumerable ones. [...] The idea of being able to list or enumerate all possible texts in a language is an extremely powerful one, and it was exploited by Borel in 1927 [Tasic, 2001, Borel, 1950] in order to define a real number that can answer every possible yes/no question! You simply write this real in binary, and use the nth bit of its binary expansion to answer the nth question in French. Borel speaks about this real number ironically. He insinuates that it's illegitimate, unnatural, artificial, and that it's an "unreal" real number, one that there is no reason to believe in. Richard's paradox and Borel's number are discussed in [Borel, 1950] on the pages given in the list of , but the next paradox was considered so important by Borel that he devoted an entire book to it. In fact, this was Borel's last book [Borel, 1952] and it was published, as I said, when Borel was 81 years old. I think that when Borel wrote this work he must have been thinking about his legacy, since this was to be his final book-length mathematical statement. The Chinese, I believe, place special value on an artist's final work, considering that in some sense it contains or captures that artist's soul. If so, [Borel, 1952] is Borel's "soul work." [...] Here it is: Borel's "inaccessible numbers:" Most reals are unnameable, with probability one. Borel's often-expressed credo is that a real number is really real only if it can be expressed, only if it can be uniquely defined, using a finite number of words. It's only real if it can be named or specifed as an individual mathematical object. [...] So, in Borel's view, most reals, with probability one, are mathematical fantasies, because there is no way to specify them uniquely. {{In Borel's view only reals that can be named belong to mathematics. Uncountability is not part of mathematics.}} Borel, E. [1950] Lecons sur la Théorie des Fonctions (Gabay, Paris) pp. 161, 275. Borel, E. [1952] Les Nombres Inaccessibles (Gauthier-Villars, Paris) p. 21. Tasic, V. [2001] Mathematics and the Roots of Postmodern Thought (Oxford University Press, New York) pp. 52, 81-82. [Gregory Chaitin: "How real are real numbers?" (2004)] http://arxiv.org/abs/math.HO/0411418

§ 231 One philosophically important way in which numbers and sets, as they are naively understood, differ is that numbers are physically instantiated in a way that sets are not. Five apples are an instance of the number 5 and a pair of shoes is an instance of the number 2, but there is nothing obvious that we can analogously point to as an instance of, say, the set {{«}}. [Nik Weaver: "Is set theory indispensable?" (2009)] http://arxiv.org/abs/0905.1680

§ 232 In June 1905 Nelson sent a letter to Hessenberg {{who invented the set that contains a certain element only if it does not contain that element}} commenting on Hilbert’s lecture "Über die Grundlagen der Arithmetik", and he expressed his disappointment about Hilbert’s ideas. Rather perplexed he wrote: “To remove the contradictions in set theory, he [i.e., Hilbert] intends to reform (not set theory but) logic. Well, we shall see, how he will do it.” Hessenberg answered quite to the point: "I do not at all consider it as paradoxical that one has to reform logic in order to make set theory free of contradictions. First of all it is not yet possible to separate logic sharply from arithmetical considerations. Secondly, however: If there are paradoxes in set theory, then either the are not correct or the concepts generated are contradictory." {{Trust in infallibility of set theory has been always well formed.}} In both cases, Hessenberg continued, it is a logical task to uncover the mistakes. According to the laws of logic a thing a falls under the concept b or not. No other principle is needed for the concept of a set. {{That is not always the case. An infinite path cannot be localized in the Binary Tree although each of its nodes can be localized there. Whether or not the path belongs to the Binary Tree depents on the intentions of the path finder (scout).}} Hessenberg stressed that Hilbert very much strengthened the requirements for building concepts in order to avoid the resulting paradoxes. [Volker Peckhaus: "Paradoxes in Göttingen", p. 10f] http://kw.uni-paderborn.de/fileadmin/kw/institute- einrichtungen/humanwissenschaften/philosophie/personal/peckhaus/Texte_zum_Download/pg.p df

§ 233 The set of all termination decimals is a subset of –. If the set of all terminating decimals of the unit interval is arranged as set of all terminating paths of the decimal tree, unavoidably all irrationals are written as infinite paths too. But we know that it is impossible to write the path of even one single irrational number, let alone of several or infinitely many or uncountably many. So belief in the above requires strong faith. A view without faith is this: There is no irrational path at all. But that would destroy the pet dogma of matheology, namely uncountability. The question is, how come uncountably many irrational paths into being during the countable process of constructing the complete decimal tree by constructing all its countably many nodes. Provably none of the irrationals is constructed in any step. And an additional question for skilled matheologians: If we delete all paths containing digits 2, 3, 4, 5, 6, 7, 8, and 9 from the decimal tree of finite paths: Do all irrationals remain? Is infinity the only necessary condition of matheological belief?

§ 234 Enumerate all rational numbers to construct a Cantor-list. Replace the diagonal digits ann by dn in the usual way to obtain the anti-diagonal d. Beyond the n-th line there are f(n) rational numbers the first n digits of which are the same as the first n digits d1, d2, d3, ..., dn of the anti- diagonal. " n in Ù: f(n) > k for every k in Ù. Define for every n in Ù the function g(n) = 1/f(n) = 0. In analysis the limit of this function is limnض g(n) = 0. So matheology with its limit limnض f(n) = 0 is incompatible with analysis. Since analysis is a branch of mathematics, matheology is incompatible with mathematics.

§ 235 Rough set theory has an overlap with many other theories dealing with imperfect knowledge {{a similarity that it shares with transfinite set theory}}. [Zdzisław Pawlak: "Rough set theory and its applications", J. Telecomm. Information Theory (3/2002) 7-9] http://www.nit.eu/czasopisma/JTIT/2002/3/7.pdf

§ 236 Tarski’s theorem: (For all infinite sets X there exists a bijection of X to XäX) fl (Axiom of Choice). [...] Fréchet and Lebesgue refused to present it. Fréchet wrote that an implication between two well known propositions is not a new result. Lebesgue wrote that an implication between two false propositions is of no interest. [Jan Mycielski: "A System of Axioms of Set Theory for the Rationalists", Notices of the AMS 53,2 (2006) 206-213] http://www.ams.org/notices/200602/fea-mycielski.pdf

§ 237 The Cantor's set theory is a Trojan Horse of the mathematics-XX: on the one hand, it is a natural, visual, universal language to describe mathematical objects, their properties and relations, originating from the famous Euler's "logical circles", and just therefore this language was accepted by all mathematicians with a natural enthusiasm. However, on the other hand, together with the language, Cantor's transfinite conceptions and constructions (like the actualization of all infinite sets, a distinguishing of infinite sets by the number of their elements (i.e., their cardinalities), the hierarchy of ordinal and cardinal transfinite numbers, continuum hypothesis, etc.) went into the mathematics-XX. Just the Cantor's actualization of infinite sets generated a lot of set-theoretical paradoxes and, ultimately, the Third Great Crisis in foundations of mathematics in the beginning of the XX c. The theme itself of the present conference shows that the problem of the actual infinity is not closed and the Third Great Crisis in foundations of mathematics goes on hitherto. [A.A. Zenkin: "Scientific Intuition of Genii Against Mytho-'Logic' of Cantor‘s Transfinite Paradise" Procs. of the International Symposium on “Philosophical Insights into Logic and Mathematics,” Nancy, France, 2002, p. 1] http://www.ccas.ru/alexzen/papers/CANTOR-2003/Zenkin%20PILM2002.doc

§ 238 Sir, I just came across your paper on "Cantor's Theorem" that there is no bijection from a set to its power set. I think you are right about the set M of "non-generators" being paradoxical. [...] I have been troubled about set theory since they told me in school that there is a rational between every two irrationals, yet more irrationals than rationals. It is obvious that "between every two irrationals there is a rational" implies that there are as many rationals as irrationals. However, I am frustrated that this could be so hard to prove while being immediately obvious to the intuition. I am also convinced that there cannot be more distinct Dedekind cuts than distinct rational numbers. Just drawing a sketch of some Dedekind cuts convinces me. The Dedekind cuts are 1) nonempty and 2) totally ordered by the relation "is a proper subset of". For finite sets it is easy to see, and prove by induction, that for such a collection of sets there are no more sets than elements. But I do not know how to make this a transfinite induction. Thank you for reading my long email. I hope that people are listening to your arguments!

Dear NN, You were right. The reason is: There are at most countably many finite definitions like e = Σ1/n!. That is undisputed. So if there should be uncountably many reals, most of them cannot be defined - or can only be defined by infinite sequences. But that means the same as being undefined, because none of those sequences defines a number unless you know the last digit - which is impossible. So those "reals" cannot be used in mathematics (which means communication) because they cannot be communicated. They are not really real. And here comes a simple proof that the notion of uncountablility is in fact nonsense: Construct all real numbers of the unit interval as infinite paths of the complete infinite Binary Tree. It contains all real numbers between 0 and 1 as infinite paths i. e. infinite sequences of bits. [...] The complete tree contains all infinite paths. The structure of the Binary Tree excludes that are any two initial segments, Bk and Bk+1, which differ by more than one infinite path. (In fact no Bk does contain any infinite path - but that is not important for the argument.) Hence either there are only countably many infinite paths. Or uncountably many infinite paths come into the tree after all finite steps of the sequence have been done. But if so, then it is by far more probable to assume that the single Cantor-diagonal comes into the Cantor-list after all lines at finite places have been searched. And then we have no reason to assume the existence of uncountable sets.

§ 239 We have, it is true, a clear idea of division, as often as we think of it; but thereby we have no more a clear idea of infinite parts in matter, than we have a clear idea of an infinite number, by being able still to add new numbers to any assigned numbers we have: endless divisibility giving us no more a clear and distinct idea of actually infinite parts, than endless addibility (if I may so speak) gives us a clear and distinct idea of an actually infinite number; they both being only in a power still of increasing the number, be it already as great as it will. So that of what remains to be added (wherein consists the infinity) we have but an obscure, imperfect, and confused idea, from or about which we can argue or reason with no certainty or clearness, no more than we can in arithmetic, about a number of which we have no such distinct idea as we have of 4 or 100; but only this relative obscure one, that compared to any other, it is still bigger: and we have no more a clear positive idea of it when we say or conceive it is bigger, or more than 400,000,000, than if we should say it is bigger than 40, or 4; 400,000,000 having no nearer a proportion to the end of addition, or number, than 4. For he that adds only 4 to 4, and so proceeds, shall as soon come to the end of all addition, as he that adds 400,000,000 to 400,000,000. And so likewise in eternity, he that has an idea of but four years, has as much a positive complete idea of eternity, as he that has one of 400,000,000 of years: for what remains of eternity beyond either of these two numbers of years is as clear to the one as the other; i. e. neither of them has any clear positive idea of it at all. For he that adds only four years to 4, and so on, shall as soon reach eternity, as he that adds 400,000,000 of years, and so on; or, if he please, doubles the increase as often as he will: the remaining abyss being still as far beyond the end of all these progressions, as it is from the length of a day or an hour. For nothing finite bears any proportion to infinite; and therefore our ideas, which are all finite, cannot bear any. Thus it is also in our idea of extension, when we increase it by addition, as well as when we diminish it by division, and would enlarge our thoughts to infinite space. After a few doublings of those ideas of extension, which are the largest we are accustomed to have, we lose the clear distinct idea of that space: it becomes a confusedly great one, with a surplus of still greater; about which, when we would argue or reason, we shall always find ourselves at a loss; confused ideas in our arguings and deductions from that part of them which is confused always leading us into confusion. [J. Locke: "The Works of in Nine Volumes", 12th ed., Vol. 1. Chapter XXIX, §16: "Of Clear and Obscure, Distinct and Confused Ideas", Rivington, London (1824)] http://oll.libertyfund.org/title/761/80774/1923786

§ 240 Consider a Cantor-list that contains a complete sequence (qk) of all rational numbers qk. The first n digits of the anti-diagonal d are d1, d2, d3, ..., dn. It can be shown for every n that the Cantor-list beyond line n contains infinitely many rational numbers qk that have the same sequence of first n digits as the anti-diagonal d. Proof: There are infinitely many rationals qk with this property. All are in the list by definition. At most n of them are in the first n lines of the list. Infinitely many must exist in the remaining part of the list. So we have obtained: " n $ k: d1, d2, d3, ..., dn = qk1, qk2, qk3, ..., qkn This theorem it is not less important than Cantor's theorem: For all " k: d ∫ qk Both theorems contradict each other with the result that finished infinity as presumed for transfinite set theory is not a valid mathematical notion.

§ 241 The aim of the production was to find visual and theatrical ways of expressing the idea of infinity. The audience did learn some math, but the main impact was at the experiential level.

The scenes each concerned aspects of infinity. The first showed Hilbert's hotel. This is a hotel with an infinite amount of rooms. Even if each room is occupied, it can accommodate a new guest: each of the present guests move one room along the line. This does not make life easier for the hotel owner, but is clearly possible given the concept of infinity – which may nonetheless be too complicated for efficient hotel management! The hotel was seen as endless doors lining the wall stretching into the rafters of the warehouse.

Another scene deals with eternal life. The audience sees very, very old people in wheelchairs or under hairdryers, reading and trying to pass the time. The surroundings are black and enclosed, so that a stifled, monotonous atmosphere is created. Barrow describes it thus: "It makes us think about living forever, exploring the social, religious and human implications of infinite life for everything from life insurance, how to set punishment for crime and recompense for negligence when an infinite future is taken away, and what to make of religion that promises everlasting life. […] The action takes place mostly above the audience with old chrones conveyed in chairs on monorails."

The third scene takes place in a large space full of corridors with mirrors at the end, dramatizing Jorge Luis Borges' parable of the library of Babel. The audience wanders through the corridors which are filled with empty bookcases, and the actors around them are identically masked and clothed, repeating the same words, seemingly an endless amount of identical people. The audience feels they are wandering through an infinite universe where anything that can happen will happen. {{Can and will it happen also that everywhere nothing happens?}}

The fourth scene (finally, the mathematicians will sigh) brings us Georg Cantor himself, the father of our modern concept of infinity. Set in a hospital, it dramatizes Cantor's conflict with Kronecker about the nature of infinity. Cantor, covered with bandages, sits in a wheelchair as Kronecker rants at him. {{This stresses that the author of the play did not know the historical facts.}}

[Judy Kupferman: Infinity in Theatre] http://www.jewish-theatre.com/visitor/article_display.aspx?articleID=579

§ 242 We should not say that the least number not definable in less than 19 words is 'definable' in less than 19 words [...] Of course one could replace 'definable' in the phrase with a bare 'referrable to' and then it might seem that the paradox would reappear in another guise: the least number not referrable to in 19 words is clearly referrable to in less than 19 words. But now Donnellan's Distinction comes into its own: for there is no paradox in the man with martini in his glass having no martini in his glass - once one appreciates the difference between reference and attribution. [H. Slater: "The Uniform Solution of the Paradoxes" (2004)] http://msc.uwa.edu.au/philosophy/about/staff/hartley_slater/publications/the_uniform_solution_of _the_paradoxes

§ 243 So any reasonably complete account of what mathematics is, or what mathematical activity is, must ultimately confront the issue of what mathematicians are trying to accomplish - at least if it is to be relevant to actual mathematical activity. This is very difficult to get a hold of, especially in light of the fact that mathematicians are not in anything like full agreement as to what they are trying to accomplish. What makes matters more difficult still, is that writing about "what is mathematics, and what are we trying to accomplish" is not considered normal professional activity among mathematicians. This statement is not so negative. After all, "what is philosophy, and what are we trying to accomplish" is rarely a topic of the leading philosopher's papers either. I gather that philosophers did not like it when Rorty wrote about this. I asked Kripke if he would write about this and related matters, and he made it clear that he wouldn't touch it with a xxxxxxxx foot pole! {{The length is given in the unary system. To be absolutely safe?}} [...] I do get the feeling that this situation is better in physics. This may be partly due to the apparent fact(?) that it is relatively clear from the outset "what physics is and what physicists are trying to accomplish" than "what mathematics is and what mathematicians are trying to accomplish". [Harvey M. Friedman: "Philosophical Problems in Logic" (2002)] http://www.math.ohio-state.edu/~friedman/manuscripts.html http://www.math.ohio-state.edu/~friedman/pdf/Princeton532.pdf

A flimsy attempt to mystify mathematics. Algebradabra! And yet it is so simple. Mathematicians calculate in order to obtain correct results, i.e., results which agree with reality. If the experiments are complicated and some physics is required in addition, we call it applied or im- pure mathematics. If the experiments are simple enough to be carried out with marbles, building blocks or strings, we call it pure or dis-plied mathematics. Both kinds of mathematics are housed under the roof of the faculty of science. And Cantor himself has done quite a lot of im-pure mathematics, namely his lessons on mechanics: Auch die drei Wintervorlesungen: analytische Mechanik [...] werden mich von den anderen Gebieten fern halten. [Cantor to Mittag-Leffler, 6 Sept. 1885] [...] und so fange ich denn morgen 4. Mai meine angekündigte 5 stündige Vorlesung über analyt. Mechanik an. [Cantor to Jourdain, 3 May 1905] His last lesson in winter semester 1910/11 was on analytical mechanics {{my translation}}. [W. Purkert, H.J. Ilgauds: "Georg Cantor 1845-1918", Birkhäuser, Basel (1987) p. 165] By the way: Lectures in General Sciences are not an invention of Bavarian Universities of Applied Sciences (laudably they are obligatory here, in contrast to other countries and nations). Even with over 70 Cantor planned lessons, which unfortunately could not be realized. In his estate an announcement of 3 May 1917 has been found: In this summer semester I intend to read privatim Aristotelian logic for all four faculties, Wednesday and Saturday 9 to 10 o'clock {{my translation}}. [loc. cit.]

§ 244 Consider the diagonal d = d1, d2, d3, ... of a Cantor list, constructed by some appropriate digit-substitution dn ∫ ann. For instance, if ann > 5 let dn = 2, and if ann < 6 let dn = 8.

If we have: " n: dn is an even digit. Can it be then, that d contains any odd digit?

If we have: " n: d1, d2, d3, ... dn the first n digits are even. Can it be then, that d contains any odd digit?

If we can prove: " n: dn has property P. Can it be then that there is a digit with the property ŸP?

If we can prove: " n: d1, d2, d3, ... dn have the property P. Can it be then that there is a digit with the property ŸP?

Cantor speaks of the Inbegriff (set) of all positive numbers n, which should be denoted by the symbol (n). Does that differ from what today is denoted by Ù or "all n œ Ù"?

Zermelo requires a set that with a also contains {a}. What is the difference to an inductive set that can be put in bijection with Ù?

Zermelo defines identity: If M ΠN and N ΠM, then M = N.

Is diagonal d an ordered set of all digits dn such that the Inbegriff (dn) differs from {dn | n œ Ù}?

§ 245 In the present paper I would like to develop a different point of views on the continuum. [...] As a background this point-set theoretic concept is influenced by in modern civilization. 19th and 20th centuries are the centuries of individualism and the individualism played an important role in the revelation of people and high advancement of science and technology. Historically individualism came from liberalism, which in turn came from Reform of Religion by earlier Protestants, and the fundamental roots can even go upstream to Apostle Paul. Anyway by historical reason Protestantism performed an important role to the development of civilization. It is marvelous, if it is taken into consideration that religion is conservative in nature, that Protestantism contributed the advancement of science that sometimes contradicts against Bible (This is caused because Protestantism abandoned to be a religion.) [...] New point of view I am now going to propose is a "missing ring", whose trace can be seen in many part of mathematics and philosophy, and these traces and "holes" will be fulfilled by the proposal introduced below. First of all I propose that continuum is never a gathering of points and is a thing that can never be counted out by points. Continuum and point, they can co-exist but are very different concept and have no relations each other. [...] Of course we can embed numbers (points) in the continuum. By doing so we sometimes measure the length of continuum or divide continuum. But it is just embedding and not any more. Looking from the continuum embedded points exist only ideally or as an intersecting limit of stringlets. So even though you may measure continuum or do addition using continuum, it is just virtual, and what you are really doing is only arithmetical operation on conventional point-set theory. [...] As a counterpart of point-set theory string-set theory is proposed. It is asserted that the string-set is the essence of continuum in one aspect [...] And importance of introducing string-set theoretical point of view not only to make mathematics useful but also correct crippled modern civilization. {{That sounds promising.}} [Akihiko Takizawa: "String Set Theory" (2002)] http://www.geocities.co.jp/SweetHome-Ivory/6352/sub7/string.html

§ 246 Cantor's list contains real numbers r as binary or decimal fractions. Real numbers, however, are limits of binary or decimal fractions. For every terminating fraction of r, Cantor obtains a difference between r and the due terminating fraction of the anti-diagonal d: rnn ∫ dn. He concludes that this remains true for the limits of the list numbers r and d by using the argument: different sequences have different limits. But it is well known that this argument is not admissible in proofs because it is false.

§ 247 The dependence of our systems of kinds on our theories, and the dependence of these, in turn, on our interests, values, technology, and the like, make questionable, at best, the thesis that our predicates pick out real properties or natural kinds whose existence, extension, and metaphysical status are independent of any contribution of ours. And the claim that just these kinds or properties are required to answer scientific questions or provide scientific explanations supports that thesis only if backed by an account of why these questions or forms of explanation have priority - an account that does not, in turn, appeal to the practices or institutions of which they are a part, else all questions are begged. [Catherine Z. Elgin: "With Reference to Reference", Hacket Publishing Company, Inc (US) (1983) p. 35]

§ 248 Cantor's work was well received by some of the prominent mathematicians of his day, such as Richard Dedekind. But his willingness to regard infinite sets as objects to be treated in much the same way as finite sets was bitterly attacked by others, particularly Kronecker. There was no objection to a "potential infinity" in the form of an unending process, but an "actual infinity" in the form of a completed infinite set was harder to accept. [H.B. Enderton, Elements of Set Theory". Academic Press, New York (1977) p. 14f]

Glossary. Potential infinity: " n $ m : n < m For every cardinal number, there exists a greater cardinal number. Actual infinity: $ m " n : n § m There exists a cardinal number such that no cardinal number is greater. And this is only the beginning, invented for the "smallest" infinity. The whole sequence of accessible cardinal numbers is again potentially infinite. And this is only the beginning - cp. the essay about Archangels and inaccessible Cardinals: Das Kalenderblatt 1090 of 28 May 2012 (in German) http://www.hs-augsburg.de/~mueckenh/KB/KB%201001-1111.pdf

§ 249 When I was a first-year student at the Faculty of Mechanics and Mathematics of the Moscow State University, the lectures on calculus were read by the set-theoretic topologist L.A. Tumarkin, who conscientiously retold the old classical calculus course of French type in the Goursat version. [...] These facts capture the imagination so much that (even given without any proofs) they give a better and more correct idea of modern mathematics than whole volumes of the Bourbaki treatise. [...] The emotional significance of such discoveries for teaching is difficult to overestimate. It is they who teach us to search and find such wonderful phenomena of harmony of the Universe. The de-geometrisation of mathematical education and the divorce from physics sever these ties. [...] teaching ideals to students who have never seen a hypocycloid is as ridiculous as teaching addition of fractions to children who have never cut (at least mentally) a cake or an apple into equal parts. No wonder that the children will prefer to add a numerator to a numerator and a denominator to a denominator. From my French friends I heard that the tendency towards super-abstract generalizations is their traditional national trait. I do not entirely disagree that this might be a question of a hereditary disease, but I would like to underline the fact that I borrowed the cake-and-apple example from Poincaré {{who used to name a disease a disease too}}. [V.I. Arnold: "On teaching mathematics" (1997), Translated by A.V. Goryunov] http://pauli.uni-muenster.de/~munsteg/arnold.html

§ 250 Fibonacci-sequences with fatalities

The Fibonacci-sequence f(n) = f(n-1) + f(n-2) for n > 2 with f(1) = f(2) = 1 the first recursively defined sequence in human history (Leonardo of Pisa, 1170 - 1240) should be well known. A pair of rabbits that reproduces itself monthly as from the completed second month on will yield 144 pairs after the first 12 months: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.

If we assume that each pair reproduces itself after two months for the last time and dies afterwards, we get a much more trivial sequencence: 1, 1, 1, ... However the rabbits behind this numbers change. If we call them in the somewhat unimaginative but effective manner of the old Romans, we get Prima, Secunda, Tertia, Quarta, Quinta, Sexta, Septima, Octavia, Nona, Decima and so on.

A more interesting question is brought up, if the parent pair dies immediately after the birth of its second child pair. Then the births in month n can be traced back to pairs who have been born in months n-2 and n-3. g(n) = g(n-2) + g(n-3). The number f(n) of pairs in month n is given by those born in month n, i.e. g(n) and those already present in month n-1, i.e., f(n-1), minus those who died in month n (i.e. those who were born in month n-3: f(n) = g(n) + f(n-1) - g(n-3) = g(n-2) + f(n-1) g(n-2) = f(n) - f(n-1) g(n-2) = g(n-4) + g(n-5) = f(n-2) - f(n-3) + f(n-3) - f(n-4) = f(n-2) - f(n-4) For n > 4 we have with f(1) = 1, f(2) = 1, f(3) = 2, f(4) = 2. f(n) = f(n-1) + f(n-2) - f(n-4) The number of pairs during the first 12 months is 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21. The sequence grows less than the original one, but without enemys or other restrictions it will grow beyond every threshold. If we wait ¡0 days (or use the trick that the duration of pregnancy is halved in each step, facilitated by genetic evolution), we will get infinitely many pairs - a nameless number, alas of nameless rabbits, because they cannot be distinguished. The set of all Old-Roman names has been exhaused already, and even all of Peano's New-Roman names S0, SS0, SSS0, ... have been passed over to pairs which already have passed away. That is amazing, since none of the pairs of the original and much more abundant Fibonacci sequence has to miss a name.

But this sequence with fatalities can also be obtained without fatalities (killings), namely if each pair has to pause for two months after each bearth in order to breed again in the following month. Mathematically, there is no difference. Set theory, however, yields a completely different limit in this case. The limit set of living rabbits is no longer empty, but it is infinite - and every rabbit has a name.

So we obtain from set theory: The cultural assets of distinguishability of distinct objects by symbols or thoughts do not belong to the properties of Cantor's paradise. Like in the book of genesis, before Adam began to name the animals, we have a nameless paradise - but not mathematics.

§ 251 Frege, too, as Lesniewski points out, attacks those mathematicians who introduce into their theories such arbitrary 'inventions' as the empty set, merely because they prove expedient for certain purposes. Lesniewski's own strictures in this respect are directed in particular against axiomatic theories of sets such a were developed by Zermelo. These do not merely lack the sort of naturalness that would dispose one to accept them; they lack also the intrinsic intelligibility which would make their meaning clear, so that Lesniewski can in all honesty assert that he does not understand what is meant by 'set' as this term is supposed to be 'implicitly defined' by theories like Zermelo's. Lesniewski himself, in contrast, starts not from 'inventions" or from axioms or hypotheses selected for pragmatic reasons, but from what he calls intuitions, commonly accepted and meaningful to all, relating to such concepts as whole, part, totality, object, identity, and so on. The language of Lesniewski's theories is therefore an extrapolation of natural language, a making precise of what, in natural language, is left inarticulate or indistinct. [...] Hence he is mistrustful, too, of the model-theoretic semantics that has been built up on an abstract set- theoretical basis, and he is opposed also to the work of those formalist logicians who embrace an essentially abstract-algebraic approach to logic, or see logic as having to deal essentially with uninterpreted formal systems. [: "On the phases of reism" in: A. Chrudzimski, D. Lukasiewics (eds): "Actions, Products, and Things", Ontos-Verlag Frankfurt (2006) p. 124]

§ 252 The table T

1 2, 1 3, 2, 1 ... n, ..., 3, 2, 1 ... is a sequence of lines Ln, finite initial segments (1, ..., n) of the sequence of natural numbers. It contains every natural number that can be somewhere. Every number in the table T is in one line Ln and in all further lines by construction of T (always the last line is added). Every number in T is in the first column C (and in every other column too).

" n : (1, ..., n) Œ C fl (1, ..., n) œ T " n : (1, ..., n) œ T fl (1, ..., n) Œ C

Therefore it is impossible that C contains more than T and more than any line Ln of T. But we know that there is no line Ln with an actually infite set Ù of numbers (because T is a sequence of finite lines Ln). Conclusion: An actually infinite set Ù cannot be in the first column either (and nowhere else).

Same gets clear from the sets. All natural numbers can be found in each of the sets, unions of sets and sequences of sets.

{1, 2, 3, ...} = Ù {1} » {2} » {3} » ... = Ù {1}, {1, 2}, {1, 2, 3}, ... {1} U {1, 2} U {1, 2, 3} U ... = Ù {1, 1, 2, 1, 2, 3, ...} {1}, {1} U {1, 2}, {1} U {1, 2} U {1, 2, 3}, {1} U {1, 2} U {1, 2, 3} U ..., ...

Nothing of Ù is lacking in any. All of Ù is in all these sets or sequences, not only in those which are officially denoted by Ù but also in the sequences of finite sets. Therefore Ù is not actually infinite.

§ 253 Notice that if the result is a method that we do not quite recognize as mathematical, {{then the reason is that mathematics like many social standards have been perverted.}}. [...] What we have traced is a more or less simultaneous rise of pure mathematics and reevaluation of applied mathematics. Before all these, back in Newton’s or Euler’s day, the methods of mathematics and the methods of science were one and the same {{Mathematics was considered as a science. Frequently theologians like Nicole Oresme, John Wallis, Bonaventura Cavalieri or used to pursue it as an alternative to their professional occupation - today mathematics does no longer offer an alternative. Mathematics and theology have merged.}}; if the goal is to uncover the underlying structure of the world, if mathematics is simply the language of that underlying structure, then the needs of celestial mechanics (for Newton) or rational mechanics (for Euler) are the needs of mathematics. From this perspective, the correctness of a new mathematical method – say the infinitary methods of the calculus or the expanded notion of function – is established by its role in application. {{That's the philosophers (touch-)stone.}} [Penelope Maddy: "How applied mathematics became pure", Reviev Symbolic Logic 1 (2008) 16-41]

§ 254 1. Finite cannot comprehend, contain, the Infinite. - Yet an inch or minute, say, are finites, and are divisible ad infinitum, that is, their terminated division incogitable. 2. Infinite cannot be terminated or begun. - Yet eternity ab ante ends now; and eternity a post begins now. So apply to Space. 3. There cannot be two infinite maxima. - Yet eternity ab ante and a post are two infinite maxima of time. 4. Infinite maximum if cut in two, the halves cannot be each infinite, for nothing can be greater than infinite, and thus they could not be parts; nor finite, for thus two finite halves would make an infinite whole. 5. What contains infinite quantities (extensions, protensions, intensions) cannot be passed through, - come to an end. An inch, a minute, a degree contains these; ergo, &c. Take a minute. This contains an infinitude of protended quantities, which must follow one after another; but an infinite series of successive protensions can, ex termino, never be ended; ergo, &c. 6. An infinite maximum cannot but be all-inclusive. Time ab ante and a post infinite and exclusive of each other; ergo, &c. 7. An infinite number of quantities must make up either an infinite or a finite whole. I. The former. - But an inch, a minute, a degree, contain each an infinite number of quantities; therefore an inch, a minute, a degree, are each infinite wholes; which is absurd. II. The latter. - An infinite number of quantities would thus make up a finite quantity, which is equally absurd. [John Stuart Mill: "An Examination of William Hamilton’s Philosophy", The Collected Works of John Stuart Mill, Volume IX, CHAPTER XXIV: "Of Some Natural Prejudices Countenanced by Sir William Hamilton, and Some Fallacies Which He Considers Insoluble" (1865), John M. Robson (ed.), Routledge and Kegan Paul, London (1979)] http://oll.libertyfund.org/?option=com_staticxt&staticfile=show.php%3Ftitle=240&chapter=40898 &layout=html#a_761210

§ 255 Let S = (1), (1, 2), (1, 2, 3), ... be a sequence of all finite initial sets si = (1, 2, 3, ..., i) of natural numbers.

Every natural number n is in some term si of S: »si = Ù. (0) " n $ i: n e si.

S contains si+1 after si. So we have (1) " n " i : (n § i ñ n œ si) ⁄ (n > i ñ n – si).

There is no term si of S that contains all natural numbers. This condition requires that in every term at least one natural number is missing. (2) $ j, k, m, n : m œ sj ⁄ m – sk ⁄ n – sj ⁄ n œ sk.

(2) is in contradiction with (1).

§ 256 In his dissertation of 1907, Brouwer had actually explained how he could accept some of Cantor’s ideas, including his transfinite numbers ω, ω + 1, … up to a certain point (as long as they are denumerable and in a certain sense constructible {{i.e., given by a finite formula or rule}}) but not the further concepts of a totality of all such denumerable numbers.[...]. And it was not the set-theoretic paradoxes that caused his reaction. As he remarked in 1923, an incorrect theory, even if it cannot be checked by any contradiction that would refute it, is none the less incorrect, just as a criminal policy is none the less criminal even if it cannot be checked by any court that would curb it. [...] The point for the intuitionists is that mathematics is a mental construction erected freely by the mind. It is simply an illusion to conceive of mathematics as dealing with independently existing objects, with an objective reality somehow external to the mind. {{"God created man in His own image? Rather man created God in his. [Georg Christoph Lichtenberg, Göttingen]. If there are Gods, then only such that are man-made. And if there are numbers, then only such that are man-made. Mental constructions cannot exist without mind - even if matheologians are prepared to prove the contrary (double meaning intended).}} But this is what modern mathematics does: the objects of the theory are conceived as elements of a totality or set that is regarded as given, totally independently of the thinking subject. This feature is deeply embedded in the methods employed in mathematics, and (following Bernays, a key collaborator of Hilbert) it is often called the “Platonism” of modern mathematics.

{{Nothing is so out of date as the utopies of yesterday: 1984, 2001, transfinite set theory.}}

Meanwhile, the constructivists’ treatment of mathematics – exemplified by intuitionism – is based on careful consideration of the processes by which numbers, etc., are defined or constructed. Each and every thing that a mathematician can legitimately talk about must have been explicitly constructed in a mental activity.

As time went by, Brouwer realized that it was better to avoid talking of “sets” at all, and he introduced new terminology (“species” and “spreads”). [...] As Brouwer’s reconstruction of mathematics developed in the 1920s, it became more and more clear that intuitionistic analysis was extremely subtle, complicated and foreign. Brouwer was not worried, for “the spheres of truth are less transparent than those of illusion,” as he remarked in 1933.

[José Ferreirós: "Paradise Recovered? Some Thoughts on Mengenlehre and Modernism", (2008)]

§ 257 The set of all possible computer programs is countable {{if going into concrete details of computers, we can even say more restrictively: The number of all possible programs is finite and certainly less than 210100. But of course Turing did not refer to real computers and did not know anything about the memory space of an exploitable surronding. Therefore let us analyze his approach here.}}, therefore the set of all computable reals is countable, and diagonalizing over the computable reals immediately yields an uncomputable real. Q.E.D. {{Well, it is not always that easy to diagonalize over finite sequences. The following list has no diagonal: 0 1 So we must take some more care, as will be done in the following.}} Let's do it again more carefully. Make a list of all possible computer programs. Order the programs by their size, and within those of the same size, order them alphabetically. The easiest thing to do is to include all the possible character strings that can be formed from the finite alphabet of the programming language, even though most of these will be syntactically invalid programs. Here's how we define the uncomputable diagonal number 0 < r < 1. Consider the kth program in our list. If it is syntactically invalid, or if the kth program never outputs a kth digit, or if the kth digit output by the kth program isn't a 3, pick 3 as the kth digit of r. Otherwise, if the kth digit output by the kth program is a 3, pick 4 as the kth digit of r. This r cannot be computable, because its kth digit is different from the kth digit of the real number that is computed by the kth program, if there is one. Therefore there are uncomputable reals, real numbers that cannot be calculated digit by digit by any computer program. [...] In other words, the probability of a real's being computable is zero, and the probability that it's uncomputable is one. [Who should be credited for this measure-theoretic proof that there are uncomputable reals? I have no idea. It seems to have always been part of my mental baggage.] {{The error occurs always at the same place - that is common to all "uncountability-proofs". The completed infinite number of all finite programs is presupposed. Otherwise the just defined number would be generated by another finite process with a finite program. But the set of all finite things is not a completed infinity, it is potentially infinite, it is not a set and does not contain all progarms but only every finite program up to every length. After all you cannot use a program that is longer than the longest program that is used. Nevertheless, up to every length, only a non-measurably small share of all usable finite programs has bee used.}} In spite of the fact that most individual real numbers will forever escape us, the notion of an arbitrary real has beautiful mathematical properties and is a concept that helps us to organize and understand the real world. Individual concepts in a theory do not need to have concrete meaning on their own; it is enough if the theory as a whole can be compared with the results of experiments. - So much for mathematics! {{Mathematics?}} [Gregory Chaitin: "How real are real numbers?" (2004)] http://arxiv.org/abs/math.HO/0411418

Some mathematicians write so beautifully that they should be poets. I mean, they should be poets instead of being mathematicians. As poets, they wouldn't be doing any real harm. Artistic license is out of place in mathematics. [David Petry, sci.math, Matheology § 257, 22 April 2013] http://groups.google.com/group/sci.math/browse_frm/thread/f7249ecd519bcfde?scoring=d&

§ 258 So what about Cantor’s much celebrated non-denumerable real? Where is it? Did Cantor produce such a real number? No, he merely sketched out the logic for a nonterminal procedure that would produce an infinitely long digit string representing a real number that would not be in the input stream of enumerated reals. Cantor’s procedure, and with it his celebrated nondenumerable, infinitely long real number, will appear with 100% certainty in the denumerable list of procedures. {{That's the point: Every diagonal number can be distinguished at a finite position from every other number. But if all strings are there to any finite dephts, as is easily visualized in the Binary Tree, then there is no chance for distinction at a finite position - and other positions are not available.}} There is no non-denumerable real, and every source of real numbers is denumerable [...] Implications throughout mathematics that build upon Cantor’s Diagonal Proof must now be carefully reconsidered. So Who Won? Professor Leopold Kronecker was right. Irrationals are not real {{ - at least they have no real strings of digits, and only countably many of them can be defined in a language that can be spoken, learned and understood}}. God made all the integers and Man made all the rest {{and in addition something more - unfortunately.}} [Brian L. Crissey: "Kronecker 1, Cantor 0: The End of a Hundred Years’ War"] http://www.briancrissey.info/files/Kronecker1Cantor0.pdf

§ 259 A discussion in sci.logic*) yielded the following remarkable results with respect to a list having infinitely many lines

{1} {1} U {2} {1} U {2} U {3} ...

Each line contains as many unioned sets as its line number indicates but does not contain a line Ù, since each line has a finite last number n.

On the other hand, there are infinitely many lines and, as each line adds one natural number, there are infinitely many natural numbers in the list. Since, by construction, every finite initial segment sn = {1, 2, 3, ..., n} is in one single line, all finite initial segments are in one single line. But Ù is not more than all its finite initial segments.

Otherwise there must exist at least two finite initial segments such that

$ j, k, m, n : m œ sj ⁄ m – sk ⁄ n – sj ⁄ n œ sk.

Further, if all lines of the list are written within one single line then Ù is in this single line.

Further if the list is prepared such that (for n > 1) after adding line sn the preceding line sn-1 is removed, then the list, again consisting of a single line only, but by construction never being empty, is empty.

[*) Matheology § 255, sci.logic, April 2013] http://groups.google.com/group/sci.logic/browse_frm/thread/83ff0cf1d8f6e48a?scoring=d&

§ 260 Potential Infinity: " n $ m : m > n

For every string of the list

0.1 0.11 0.111 ... there is a longer one. This, however, does not prove the uncountability of strings, because every longer string is in the list too.

A proof of uncountability needs a string that differs from every string of the list and, by construction of the list, is longer than all. That is called

Actual Infinity: $ m " n : m > n

But it is impossible to construct such a string because, by definition, all digits 1 at natural indices are already in the strings of the list. Therefore Cantor's argument fails to produce a string different from all strings of the list at natural indices.

Example: There is no irrational number that can be distinguished by any sequence of its digits from all rational numbers. For that sake always a finite definition is required. But finite definitions cannot result from Cantor's argument.

§ 261 [...] when I found it, I thought in the beginning that it causes invincible problems for set theory that would finally lead to the latter’s eventual failure; now I firmly believe, however, that everything essential can be kept after a revision of the foundations, as always in science up to now {{of course, just like the teachings of world spirit, geocentric system, phlogiston theory, ether, or principle of causality}}. I have not published this contradiction {{what a bad word escaped Hilbert's mouth}} [D. Hilbert: "Logische Principien des mathematischen Denkens, lecture course in the summer term 1905, lecture notes by Ernst Hellinger", Library of the Mathematics Seminar of the University of Göttingen, p. 204] [...] The paradox is based on a special notion of set which Hilbert introduces by means of two set formation principles starting from the natural numbers. The first principle is the addition principle. In analogy to the finite case, Hilbert argued that the principle can be used for uniting two sets together “into a new conceptual unit [...], a new set that contains each element of either sets.” This operation can be extended: “In the same way, we are able to unite several sets and even infinitely many into a union.” The second principle is called the mapping principle. Given a set M, he introduces the set MM of self-mappings of M to itself. [Hilbert used the German term “Selbstbelegung” which is translated here by “selfmapping”.] A self-mapping is just a total function which maps the elements of M to elements of M. [In classical logic, MM is isomorphic to 2M, and the set of all mappings from M to {0, 1} is isomorphic to P(M), the power set of M.]

Now, he considers all sets which result from the natural numbers “by applying the operations of addition and self-mapping an arbitrary number of times.” By use of the addition principle which allows to build the union of arbitrary sets one can “unite them all into a sum set U which is well- defined.” In the next step the mapping principle is applied to U, and we get F = UU as the set of all self-mappings of U. Since F was built from the natural numbers by using the two principles only, Hilbert concludes that it has to be contained in U. From this fact he derives a contradiction. Since “there are ‘not more’ elements” in F than in U there is an assignment of the elements ui of U to elements fi of F such that all elements of fi are used. Now one can define a self-mapping g of U which differs from all fi. Thus, g is not contained in F. Since F was assumed to contain all selfmappings we have a contradiction. In order to define g Hilbert used Cantor’s diagonalization method. [...] Hilbert finishes his argument with the following observation: "We could also formulate this contradiction so that, according to the last consideration, the set UU is always bigger [of greater cardinality] than U but, according to the former, is an element of U." [...] Considering Cantor’s general definition of a set as the comprehension of certain well- distinguished objects of our intuition or our thinking as a whole, one can justly ask whether the sets of all cardinals, of all ordinals or the universal set of all sets are sets according to this definition, i. e., whether an unrestricted comprehension is possible. Cantor denies this. Hilbert, on the other hand, introduces two alternative set formation principles, the addition principle and the mapping principle, but they lead to paradoxes as well. In avoiding concepts from transfinite arithmetic Hilbert believes that the purely mathematical nature of his paradox is guaranteed. For him, this paradox appears to be much more serious for mathematics than Cantor’s, because it concerns an operation that is part of everyday practice of working mathematicians. [Volker Peckhaus: "Paradoxes in Göttingen" (2003)] {{This headline is written in English. But it acquires a good meaning when understood in German. - Unfortunately no longer online.}} http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.163.4718

§ 262 As an interesting detail let us add that in response to the paper (1927) by von Neumann it was reacted critically by Stanisław Leśniewski who published the paper “Grundzüge eines neuen Systems der Grundlagen der Mathematik” (1929) in which he critically analyzed various attempts to formalize logic and mathematics. Leśniewski among others expresses there his doubts concerning the meaning and significance of von Neumann’s proof of the consistency of (a fragment of) arithmetic and constructs – to maintain his thesis – a “counterexample”, namely he deduce (on the basis of von Neumann’s system) two formulas a and Ÿa, hence an inconsistency. Von Neumann answered to Leśniewski’s objections in the paper “Bemerkungen zu den Ausführungen von Herrn St. Leśniewski über meine Arbeit ‘Zur Hilbertschen Beweistheorie’”(1931). Analyzing the objections of Leśniewski he came to the conclusion that there is in fact a misunderstanding resulting from various ways in which they both understand principles of formalization. {{No contradictions! Never! None!}} He used also the occasion to fulfill the gap in his paper (1927). Add also that looking for a proof of the consistency of the classical mathematics and being (still) convinced of the possibility of finding such a proof (in particular a proof of the consistency of the theory of real numbers) von Neumann doubted whether there are any chances to find such a proof for the set theory – cf. his paper (1929). [Roman Murawski: "John von Neumann and Hilbert's school of foundations of mathematics", Studies in logic, grammar and rhetoric 7,20 (2004) p. 49] http://www.google.de/url?sa=t&rct=j&q=&esrc=s&frm=1&source=web&cd=1&sqi=2&ved=0CDIQ FjAA&url=http%3A%2F%2Flogika.uwb.edu.pl%2Fstudies%2Fdownload.php%3Fvolid%3D20%2 6artid%3Drm&ei=rBpsUf2eDsKqtAbk2oGQCA&usg=AFQjCNE1Em4kIz3pFqzC8wtfGwUoALATI A&sig2=EwLxn6T5JatAx-ik-QyzAA&bvm=bv.45175338,d.Yms

§ 263 There are things. They exist by energy (Joule). There are ideas. They exist by information (bit).

If a plumber asserts to have twenty thousand hammers in his toolbag, I would carefully supervise his work, if done in in my house.

If a farmer asserts that his barn contains twenty billions potatoes, I will buy from him expecting very big potatoes. *)

If a mathematician asserts that his model contains uncountably many indistinguishable but distinct numbers, this will not cause a sensation.

Isn't that sensational?

*) The joke is based on the German proverb fortune favours foolish farmers by increasing the diameter of their potatoes.

§ 264 Hilbert's Hotel, last chapter: checking out.

Please leave the room on the day of your departure at the latest till 11:00 a.m. Hilbert's hotel is not luxurious but expensive and notorious for frequent change of rooms. Therefore many guests prefer Math's Motel. Before checking out a guest must have occupied room number 1 (because the narrow halls are often blocked). No problem, guests are accustomed to that habit. The guest of room number 1 checks out at half past 10 and all other guests change their rooms such that all rooms remain occupied. The second guest checks out at quarter to 11. And all guests switch rooms such that no room is empty. And so on. At 11 a.m. all guests have left Hilbert's Hotel. Every room is occupied.

A fine result of set theory. It can be improved however to have a real mathematical application, enumerating the sets of rational and of irrational algebraic numbers.

First enumerate the first two rationals q2 = 1/2 and q1 = 1/3. Then take off label 1 from 1/3 and enumerate the first irrational x1. 1/3 will get remunerated and re-enumerated in the next round by label 3, when 1/2 will lose its 2 but gain label 4 instead. So 1/2 and 1/3 will become q4 and q3. Continue until you will have enumerated the first n rationals and the first n irrationals

q2n, q2n-1, ..., qn+1 and xn, xn-1, ..., x1 and if you got it by now, then go on until you will have enumerated all of them. Then you have proved in ZFC that there are no rational numbers. (If you like you can also prove that there are no algebraic irrational numbers. But that's not a contradiction, of course.)

§ 265 Abstract. This paper examines the possibilities of extending Cantor’s two arguments on the uncountable nature of the set of real numbers to one of its proper denumerable subsets: the set of rational numbers. The paper proves that, unless certain restrictive conditions are satisfied, both extensions are possible. It is therefore indispensable to prove that those conditions are in fact satisfied in Cantor’s theory of transfinite sets. Otherwise that theory would be inconsistent. [...] We have just proved [...] the alternatives of Cantor's 1874-argument on the cardinality of the real numbers can be applied to the set – of rational numbers, except the last one, that applies only if the common limit of the sequences of left and right endpoints of the QP-intervals is rational. Evidently, if Cantor’s 1874-argument could be extended to the rational numbers we would have a contradiction: the set – would and would not be denumerable. Accordingly, in order to ensure the impossibility of that contradiction, each of the following points have to be proved: Whatsoever be the rational interval (a, b) and whatsoever be the reordering of , it must hold: (1) The number of QP-intervals can never be finite. (2) The sequences of endpoints and can never have different limits. (3) The common limit of and can never be rational. [...] Until those proofs be given, Cantor’s 1874-argument should be suspended, and the possibility of a contradiction involving the foundation of (infinitist) set theory should be considered. [Antonio Leon Sanchez: "Cantor versus Cantor" (2010)] http://arxiv.org/PS_cache/arxiv/pdf/1001/1001.2874v3.pdf

On the other hand, the proof can feign the uncountability of a countable set. If, for instance, the alternating harmonic sequence

n ωn = (-1) /n Ø 0 is taken [...], yielding the intervals (-1, 1/2), (-1/3, 1/4), ... we find that its limit 0 does not belong to the sequence, although the set of numbers involved, Ù » {0}, is obviously denumerable [...] The alternating harmonic sequence does not, of course, contain all real numbers, but this simple example demonstrates that Cantor's first proof is not conclusive. Based upon this proof alone, the uncountability of this and every other alternating convergent sequence must be claimed. Only from some other information we know their countability (as well as that of –), but how can we exclude that some other information, not yet available, in future will show the countability of ” or —? [W. Mueckenheim: "On Cantor's important proofs" (2003)] http://arxiv.org/pdf/math.GM/0306200

§ 266 For every natural number n the sequence (10-1, 10-2, 10-3, ..., 10-n) can be reflected at the 100-position to (10n, ..., 103, 102, 101). Obviously this transformation does not depend on the number of terms and does not depend on the number of exponents, but solely on the condition that all exponents are natural numbers. Now try to reflect the sequence for 1/9. The claim is that all exponents are natural numbers too. Has 1/9 a complete decimal representation with only natural exponents?

§ 267 ... our axioms [of set theory], if interpreted as meaningful statements, necessarily presuppose a kind of Platonism, which cannot satisfy any critical mind and which does not even produce the conviction that they are consistent. [S. Feferman et al. (eds.): "Kurt Gödel, Collected Works, Vol. III, Unpublished Essays and Lectures", Oxford University Press, Oxford (1995) p. 50]

§ 268 Can a matheologian disprove the existence of matheologians?

Here are the facts: Recently a matheologian wrote: I actually question whether the existence of matheologians is consistent. As a matheologian is either a man or a woman, it follows that a matheologian is a person. Now this person believes in thoughts that nobody can think. I would surmise that thinking a thought is a prerequisite to believing that same thought. So this person (this body) thinks a thought that nobody thinks. [name withheld - from "How to distinguish between the complete and the incomplete infinite binary tree?", a question in math.stackxchange, meanwhile deleted.]

In my opinion this only shows that not all matheologians know the fundamentals of their belief. Of course the sober phrase "thinking a thought is prerequisite in believing that thought" does not hold in matheology. On the contrary, it is just the touchstone for a matheologian to believe in uncountably many thoughts, namely one for each real number as an individual, that obviously nobody can think. - Is this acceptable?

§ 269 How to distinguish a decimal representation from a set of all terminating decimal representations? It seems impossible to accomplish this task by means of one ore more digits that have finite indices, i.e., finite distances from the decimal point. It is clear that an infinite decimal representation has more digits than every finite one. But it is as clear that there are not more finite indices than all finite indices which are already used and occupied by all possible sequences of digits to produce all finite decimal representations. Is it possible to apply other tools?

§ 270 Gödel’s incompleteness results had great influence on von Neumann’s views towards the perspectives of investigations on the foundations of mathematics. He claimed that “Gödel’s result has shown the unrealizability of Hilbert’s program” and that “there is no more reason to reject intuitionism” (cf. his letter to Carnap of 6th June 1931 – see Mancosu, 1999, 39–41). He added in this letter: Therefore I consider the state of the foundational discussion in Königsberg to be outdated, for Gödel’s fundamental discoveries have brought the question to a completely different level. (I know that Gödel is much more careful in the evaluation of his results, but in my opinion on this point he does not see the connections correctly). Incompleteness results of Gödel changed the opinions cherished by von Neumann and convinced him that the programme of Hilbert cannot be realized. In the paper “The Mathematician” (1947) he wrote: My personal opinion, which is shared by many others, is, that Gödel has shown that Hilbert’s program is essentially hopeless. Another reason for the disappointment of von Neumann with the investigations in the foundations of mathematics could be the fact that he became aware of the lack of categoricity of set theory, i.e., that there exist various nonisomorphic models of the set theory. The latter fact implies that it is impossible to describe the world of mathematics in a unique way. In fact there is no absolute description, all descriptions are relative. Not only von Neumman was aware of this feature of the set theory. {{So it is good luck that set theory (including Gödel's results) is built upon finished infinity and that this notion has been proven to be contradictory. Mathematicians should celebrate this as a great triumph instead of racking their brains in order to desperately stick to matheology.}} [Roman Murawski: "John von Neumann and Hilbert's school of foundations of mathematics", Studies in logic, grammar and rhetoric 7,20 (2004) p. 50f] http://www.google.de/url?sa=t&rct=j&q=&esrc=s&frm=1&source=web&cd=1&sqi=2&ved=0CDIQ FjAA&url=http%3A%2F%2Flogika.uwb.edu.pl%2Fstudies%2Fdownload.php%3Fvolid%3D20%2 6artid%3Drm&ei=rBpsUf2eDsKqtAbk2oGQCA&usg=AFQjCNE1Em4kIz3pFqzC8wtfGwUoALATI A&sig2=EwLxn6T5JatAx-ik-QyzAA&bvm=bv.45175338,d.Yms

§ 271 Axiom of extensionality: A set X is equal to set Y if and only if they both have exactly the same elements

The union of all finite initial sequences of natural numbers F(n) = (1, 2, 3, ..., n) is equal to the union of all F without the first k F:

" k œ Ù : »nœ ÙF(n) = »nœ Ù(F(n) \ F(k)) " k œ Ù : »nœ ÙF(n) = »nœ Ù(F(n) \ »j§k F(j))

This is true because " k $ n : F(k) Õ F(n)

§ 272 In algorithmic information theory, the notion of Kolmogorov complexity is named after the famous mathematician even though it was independently discovered and published by Ray Solomonoff a year before Kolmogorov. Li and Vitanyi, in "An Introduction to Kolmogorov Complexity and Its Applications", write: Ray Solomonoff [...] introduced [what is now known as] 'Kolmogorov complexity' in a long journal paper in 1964. [...] This makes Solomonoff the first inventor and raises the question whether we should talk about Solomonoff complexity. http://en.wikipedia.org/wiki/Matthew_effect_(sociology)

The idea is that a string is random if it cannot be compressed. That is, if it has no short description. {{A string x of bits with |x| = n bit is incompressible, if no string p of bits with less than n bits exists, which defines or generates the string x (for instance via a computer program.}} Using {{Kolmogorov complexity}} C(x) we can formalize this idea via the following.

Theorem 1.2. For all n, there exists some x with |x| = n such that C(x) ¥ n. Such x are called (Kolmogorov) random.

Proof. Suppose not. Then for all x, C(x) < n. Thus for all x there exists some px such that g(px) = x and |px| < n. Obviously, if x ∫ y then px ∫ py. But there are 2n - 1 programs of length less than n, and 2n strings of length n. {{Compare the finite paths up to level n - 1 in the Binary Tree and the paths wit n nodes, i.e., those with one n-th node beyond the level n - 1}}. By the pigeonhole principle, if all strings of length n have a program shorter than n, then there must be some program that produces two different strings. Clearly this is absurd, so it must be the case that at least one string of length n has a program of length at least n. [Lance Fortnow: "Kolmogorov Complexity" (2000)] http://people.cs.uchicago.edu/~fortnow/papers/kaikoura.pdf

By the pigeonhole principle, if all of the first n natural numbers have a unary representation that is shorter then n there must be some unary representation that defines two different natural numbers. Clearly this is absurd, so it must be the case that at least one of ¡0 numbers has a unary representation of length at least ¡0.

§ 273 In Sec. 3.1 we constructed an uncomputable real r. It must be uncomputable, by construction. Nevertheless, as was the case in the Richard paradox, it would seem that we gave a procedure for calculating Turing's diagonal real r digit by digit. {{That seems only so to someone who thinks an infinite sequence could convey information.}} How can this procedure fail? What could possibly go wrong? The answer is this: The only noncomputable step has got to be determining if the kth computer program will ever output a kth digit. If we could do that, then we could certainly compute the uncomputable real r. In other words, Sec. 3.1 actually proves that there can be no algorithm for deciding if the kth computer program will ever output a kth digit. And this is a special case of what's called Turing's . In this particular case, the question is whether or not the wait for a kth digit will ever terminate. In the general case, the question is whether or not a computer program will ever halt. {{It will not if there is an infinite loop 00 Begin 10 Goto 20 20 Goto 10 30 Print "3" 40 End An infinite loop will belong to a potential infinity of programs. Infinitely many programs will never have been investigated with respect to this property. Therefore their number r will never be defined. r is and remains undefined. Therefore r is not a number. This fact has no connection to a proof of uncountability. The dyslogic of the condition "if the kth program never outputs a kth digit" becomes easily visible by slightly paraphrasing it: In case that the case never occurs. Of course that is undecidable until a numbers will be returned.}} The algorithmic unsolvability of Turing's halting problem is an extremely fundamental meta-theorem. It's a much stronger result than Gödel's famous 1931 incompleteness theorem. Why? Because in Turing's original 1936 paper he immediately points out how to derive incompleteness from the halting problem. {{Shit happens. Incompleteness happens - for instance in every potential infinity. Uncountability does not happen or exist.}} A formal axiomatic math theory (FAMT) consists of a finite set of axioms and of a finite set of rules of inference for deducing the consequences of those axioms. Viewed from a great distance, all that counts is that there is an algorithm for enumerating (or generating) all the possible theorems, all the possible consequences of the axioms, one by one, by systematically applying the rules of inference in every possible way. This is in fact what's called a breadth-first (rather than a depth-first) tree walk, the tree being the tree of all possible deductions. So, argued Turing in 1936, if there were a FAMT that always enabled you to decide whether or not a program eventually halts, there would in fact be an algorithm for doing so. You'd just run through all possible proofs until you find a proof that the program halts or you find a proof that it never halts. So uncomputability is much more fundamental than incompleteness. Incompleteness is an immediate corollary of uncomputability. But uncomputability is not a corollary of incompleteness. The concept of incompleteness does not contain the concept of uncomputability. [Gregory Chaitin: "How real are real numbers?" (2004)] http://arxiv.org/abs/math.HO/0411418

Admit only such programs which within three minutes (on a certain computer) return a number. In principle there are arbitrarily many such programs. But if you wish to construct a real number r from the output, you have to fix a last program. Alas you get only a rational number which of course can be calculated by a program. The inclusion of programs that do never halt sneaks a logical trap into that matter. In case of a not occuring case no case can be recognized (and what digit it would define).

§ 274 Mathematicians define real numbers to be an ordered field, yet when one applies Turing‘s proof that the halting problem is unsolvable to the processes that generate infinitely precise numbers (irrationals), it follows that some irrational pairs cannot be ordered without solving the halting problem and thus are not real by definition. The de facto acceptance of Cantor‘s diagonal proof means that mathematicians accept 1. infinitely precise numbers as real numbers and 2. output from nonterminal procedures as definitions of infinitely precise real numbers. Inequality of unpredictable irrationals can be detected in time, but not equality. So unpredictable irrational numbers cannot be real. [B. L. Crissey: "Unreal Irrationals: Turing Halts Cantor"] http://www.briancrissey.info/Research/Resume_files/Unreal%20Irrationals.pdf

§ 275 The term actual infinite is not a term of set theory, despite what crazy internet authors like to say. [Kwalish Kid (2006)] http://forums.philosophyforums.com/threads/actual-infinity-and-potential-infinity-23597.html

Yes, set theorists try to avoid that term in order to remain attractive to sober minds - but set theory needs actual or completed or, why not put it frankly, finished infinity and uses it heavily.

§ 276 Cantor's belief in the actual existence of the infinite of Set Theory still predominates in the mathematical world today. [A. Robinson:"The metaphysics of the calculus", in Imre Lakatos (ed.): "Problems in the philosophy of mathematics", North Holland, Amsterdam (1967) p. 39] http://www.amazon.de/Problems-Philosophy- Mathematic/dp/0444534113/ref=sr_1_cc_1?s=aps&ie=UTF8&qid=1369999937&sr=1-1- catcorr&keywords=0444534113#_

§ 277 It is clear that the theological considerations by which Cantor motivated his notion of the actual infinite, were metaphysical in nature. [A. Heyting: "Technique versus metaphysic in the calculus", in Imre Lakatos (ed.): "Problems in the philosophy of mathematics", North Holland, Amsterdam (1967) p. 43] http://www.amazon.de/Problems-Philosophy- Mathematic/dp/0444534113/ref=sr_1_cc_1?s=aps&ie=UTF8&qid=1369999937&sr=1-1- catcorr&keywords=0444534113#_

§ 278 If, for example, our set theory includes sufficient large cardinals, we might count Banach–Tarski as a good reason to model physical space [...] From this I think it is clear that considerations from applications are quite unlikely to prompt mathematicians to restrict the range of abstract structures they admit. It is just possible that as-yet-unimagined pressures from science will lead to profound expansions of the ontology of mathematics, as with Newton and Euler, but this seems considerably less likely than in the past, given that contemporary set theory is explicitly designed to be as inclusive as possible. More likely, pressures from applications will continue to influence which parts of the set-theoretic universe we attend to, as they did in the case of Dirac’s delta function; in contemporary science, for example, the needs of quantum field theory and string theory have both led to the study of new provinces of the set- theoretic universe {{with negative result. There is no meaningful application of a meaningless theory possible}}. [Penelope Maddy: "How applied mathematics became pure", Reviev Symbolic Logic 1 (2008) 16 - 41]

§ 279 Attempts to create "pure" deductive-axiomatic mathematics have led to the rejection of the scheme used in physics (observation - model - investigation of the model - conclusions - testing by observations) and its substitution by the scheme: definition - theorem - proof. It is impossible to understand an unmotivated definition but this does not stop the criminal algebraists-axiomatisators. For example, they would readily define the product of natural numbers by means of the long multiplication rule. With this the commutativity of multiplication becomes difficult to prove but it is still possible to deduce it as a theorem from the axioms. It is then possible to force poor students to learn this theorem and its proof (with the aim of raising the standing of both the science and the persons teaching it). It is obvious that such definitions and such proofs can only harm the teaching and practical work. It is only possible to understand the commutativity of multiplication by counting and re-counting soldiers by ranks and files or by calculating the area of a rectangle in the two ways. Any attempt to do without this interference by physics and reality into mathematics is sectarianism and isolationism which destroy the image of mathematics as a useful human activity in the eyes of all sensible people. [V.I. Arnold: "On teaching mathematics" (1997) Translated by A.V. Goryunov] http://pauli.uni-muenster.de/~munsteg/arnold.html

§ 280 There is a concept which is the corruptor and the dazzler of the others. I do not speak of Evil, whose limited empire is ethics; I speak of the infinite. [Jorge Luis Borges: "Los avatares de la tortuga", translated by by BombaMolotov: "The Avatars of the Tortoise"] http://bombamolotov.deviantart.com/art/The-Avatars-of-the-Tortoise-87015348

§ 281 So if actual infinities exist then there cannot be any discrete computational foundation to reality but so far no actual infinites have ever been discovered. [John Ringland] http://www.anandavala.info/TASTMOTNOR/Infinity.html

§ 282 I think, second order categoricity results are deceiving: they serve only to puzzle ordinary mathematicians who do not know enough logic to distinguish between first order and second order methods. One can say humorously, while first order reasonings are convenient for proving true mathematical theorems, second order reasonings are convenient for proving false metamathematical theorems. [L. Kalmár: "On the role of second order theories" in Imre Lakatos (ed.): "Problems in the philosophy of mathematics", North Holland, Amsterdam (1967) p. 104] http://www.amazon.de/Problems-Philosophy- Mathematic/dp/0444534113/ref=sr_1_cc_1?s=aps&ie=UTF8&qid=1369999937&sr=1-1- catcorr&keywords=0444534113#_

§ 283 One says that one quantity is the limit of another quantity, when the second can approach the first more closely than any given quantity, however small, without the quantity approaching, passing the quantity which it approaches; so that the difference between a quantity and its limit is absolutely inassignable. [...] The theory of limits is the basis of the true metaphysics of differential calculus. [...] Properly speaking, the limit never coincides, or is never equal to the quantity of which it is the limit; but it is approached more and more, and can differ by as little as one wants. The circle, for example, is the limit of the inscribed and circumscribed polygons; because it never merges with them, though they can approach it ad infinitum. This notion can serve to clarify many mathematical propositions. [Jean Le Rond d'Alembert, Jean de la Chapelle: "Limite", Encyclopédie ou Dictionnaire raisonné des sciences, des arts et des métiers, vol. 9. Paris (1765) p. 542, translated by Jeff Suzuki: "Limit", The Encyclopedia of Diderot & d'Alembert Collaborative Translation Project, Ann Arbor: MPublishing, University of Michigan Library (2012)]

§ 284 Crudely put, a potential infinite is a collection which is increasing toward infinity as a limit, but never gets there. Such a collection is really indefinite, not infinite. The sign of this sort of infinity, which is used in calculus, is ¶. An actual infinite is a collection in which the number of members really is infinite. The collection is not growing toward infinity; it is infinite, it is "complete". The sign of this sort of infinity, which is used in set theory to designate sets which have an infinite number of members, such as {1, 2, 3, ...}, is ¡0. Now [...] an actually infinite number of things cannot exist. For if an actually infinite number of things could exist, this would spawn all sorts of absurdities. [W. L. Craig: "The Existence of God and the Beginning of the Universe", Truth: A Journal of Modern Thought 3 (1991) 85-96] http://www.leaderu.com/truth/3truth11.html

§ 285 In this article, I argue that it is impossible to complete infinitely many tasks in a finite time. A key premise in my argument is that the only way to get to 0 tasks remaining is from 1 task remaining, when tasks are done 1-by-1. I suggest that the only way to deny this premise is by begging the question, that is, by assuming that supertasks are possible. 1 By definition, completing infinitely many tasks requires getting the number of tasks remaining down to 0. 2 If tasks are done 1-by-1, then the only way to get to 0 tasks is from 1 task, because if more than 1 task remains, then performing a task does not leave 0 tasks. (This reasoning holds in both the finite and infinite cases.) 3 When infinitely many tasks are attempted 1-by-1, there is no point at which 1 task remains. 4 Then from 2 and 3, there is no point at which 0 tasks remain. 5 Then from 1 and 4, it is not possible to complete infinitely many tasks. [Jeremy Gwiazda: "A proof of the impossibility of completing infinitely many tasks", Pacific Philosophical Quarterly, 93,1 (March 2012) http://onlinelibrary.wiley.com/doi/10.1111/j.1468-0114.2011.01412.x/full

Therefore it is not possible to enumerate all rational numbers (always infinitely many remain) by all natural numbers (always infinitely many remain) or to traverse the lines of a Cantor list (always infinitely many remain).

§ 286 Cantor completely contradicted the Aristotelian doctrine proscribing actual, “completed” infinities, and for his boldness he was rewarded with a lifetime of controversy, including condemnation by many of the most influential mathematicians of his time. This reaction stifled his career and may ultimately have destroyed his mental health. It also, however, gained him a prominent and respected place in the history of mathematics, for his ideas were ultimately vindicated, and they now form the very foundation of contemporary mathematics. [Math Academy Online: "You can't get there from here"] http://www.mathacademy.com/pr/minitext/infinity/

{{No, Canotor's delusions have been contradicted. Alas matheologians refuse to understand these contradictions. Here is an example:}}

Construct a Cantor-list containing all rational numbers of the unit interval. Replace the diagonal digits ann by dn in the usual way to obtain the anti-diagonal d. Beyond the n-th line there are f(n) rational numbers the first n digits of which are the same as the first n digits d1, d2, d3, ..., dn of the anti-diagonal. f(n) is infinite for every n. So we can safely say that it is possible to find at least n duplicates of d1, d2, d3, ..., dn in entries below line n. Define for every n the sequence g(n) = 1/n. g(n) has limit 0 in analysis. So in the limit there are infinitely many duplicates of the anti-diagonal d1, d2, d3, ... in the list. [user81183, June 2013] {{This shows that infinitely many numbers are in the list that up to every digit are identical to the anti-diagonal. Two numbers that are identical up to every digit are identical - in anlysis. But a student of set theory cannot risk to understand that:}} "I still don't understand the relevancy of these functions to the 'anti-diagonal'". [Asaf Karagila, June 2013] http://math.stackexchange.com/questions/412757/what-is-the-difference-between-these-limits- in-set-theory-and-analysis

§ 287 Cantor's transfinite universe became the infinite paradigm during the 20th Century. This affected educational studies, which tended to view children's responses against Cantorian ideas. Robinson's non-standard universe (Robinson, 1966) is equally authoritative (though not as well known) and it is a different paradigm. It offers researchers a release from a single paradigm and allows them to interpret children's ideas with reference to children's ideas instead of with reference to Cantorian ideas. [J. Monaghan: "Young peoples' ideas of infinity", Educational Studies in Mathematics 48,2 (2001) 239-257]

§ 288 Here are the differences in the premises which lead to differences in the results of mathematics and matheology.

Matheology requires:

1) The Binary Tree

0. / \ 0 1 / \ / \ 0 10 1 ... containing all rational numbers of the unit interval also contains all irrational numbers. If the rationals are written in the usual manner this is not the case.

2) The triangle constructed in 3-symmetry is equilateral.

d dc dac dbbc ...

If however, the triangle is constructed such that always one and the same side is expanded, then it loses 3-symmetry "in the limit".

a bb ccc ...

3) For the union of the sequence of unions of preceding sets

»({1}, {1, 2}, {1, 2, 3} , ..., {1, 2, 3, ..., n}) = {1, 2, 3, ..., n} equality holds - but not "in the limit".

In mathematics all these premises lead to different results:

1) The Binary Tree containing all rational numbers of the unit interval does not contain any irrational number (it does not even contain periodic rationals). 2) The triangle constructed in 3-symmetry is and always remains equilateral. 3) For the union of the sequence of unions of preceding sets equality holds always.

§ 289 According to Fontenelle, none of the geometers who had invented or employed the calculus of infinity had given a general theory of it; that is what he proposed to do {{in his Élémens de la géométrie de l’infini (1727)}} ... There was a great deal of discussion in the scientific community about this work, in which mathematicians found numerous paradoxes. Johann I Bernoulli, for example, in his correspondence with Fontenelle allowed his criticisms to show through his praise: he did not understand what was meant by finis indéterminables. Fontenelle attempted to defend his theory and above all his distinction between metaphysical infinity and geometric infinity: one must ignore the metaphysical difficulties in order to further geometry, and the finis indéterminables ought to be considered “as a type of hypothesis necessary until now in order to explain several phenomena of the calculus” (letter to Johann I Bernoulli, 29 June 1729). “The orders of infinite and indeterminable quantities, like the magnitudes that they represent, are only purely relative entities, hypothetical and auxiliary. ["Fontenelle, Bernard Le Bouyer (or Bovier) De", Complete Dictionary of Scientific Biography (2008) Encyclopedia.com. 15 Jun. 2013] http://www.encyclopedia.com/doc/1G2-2830901469.html

According to Cantor none of the geometers who had invented or employed the infinite had given a correct theory of it ...

§ 290 [...] the future is only a potential infinite; one can keep adding future time to it infinitely, but it is never a complete infinite set. Whereas an infinite past consists of an actual infinite number of events that have occurred. [Shrunk: "Can an actual infinite exist?", 17 Aug. 2011] http://www.rationalskepticism.org/mathematics/can-an-actual-infinite-exist-t24806.html

Therefore there is no infinite past. Somewhen it must have done the step from finity to infinity. But that step never happens.

§ 291 Only someone who (like the intuitionist) denies that the concepts and axioms of classical set theory have any meaning (or any well-defined meaning) could be satisfied with such a solution {{undecidability of the continuum hypothesis}}, not someone who believes them to describe some well-determined reality. For this reality Cantor's conjecture must be either true or false, and its undecidability from the axioms known today can only mean that these axioms do not contain a complete description of this reality; [...] not one plausible proposition is known which would imply the continuum hypothesis. Therefore one may on good reason suspect that the role of the continuum problem in set theory will be this, that it will finally lead to the discovery of new axioms which will make it possible to disprove Cantor's conjecture. [Kurt Godel: "What is Cantor's Continuum Problem?", The American Mathematical Monthly, 54,9 (1947) p. 520, 524] http://www.personal.psu.edu/ecb5/Courses/M475W/Readings/Week06- IntoTheTwentiethCentury-10- 8/Supplementary/What%20is%20Cantor's%20Continuum%20Problem,%20by%20Kurt%20God el.pdf

§ 292 Ultrafinitism does not mean confining mathematics to a segment of the natural numbers, or to a particular hereditarily finite set [...] Instead, ultrafinitism looks for nonclassical objects, and it looks to nature -- the world of natural appearances or phenomena -- for them. [Robert Tragesser: "FOM: Part I: Ultrafinitism, Naturalism, Vagueness", 10 April 1998] http://www.cs.nyu.edu/pipermail/fom/1998-April/001825.html

§ 293 [...] parallel considerations would force us to conclude, not merely that a series of discrete, successive events must have a first member, but also that such a series must have a final member. Anyone who thinks that an end-less series of events is possible must therefore reject this popular line of argument against the possibility of an actual infinite. [Wes Morriston: "Beginningless Past, Endless Future, and the Actual Infinite"] http://www.academia.edu/647482/Beginningless_Past_Endless_Future_and_the_Actual_Infinite

Endless for ever or endless at the moment are two different things. The time passed since the big bang is not endless. The time to pass may be endless. But the passed time will never report "infinity reached".

729 § 294 Does the Bernays' number 67257 actually belong to every set which contains 0 and is closed under the successor function? The conventional answer is yes but we have seen that there is a very large element of fantasy in conventional mathematics which one may accept if one finds it pleasant, but which one could equally sensibly (perhaps more sensibly) reject. [R. Parikh: "Existence and feasibility in arithmetic", Journal of Symbolic Logic 36 (1971) 494-508]

§ 295 When God, at the end of all time, will check what of his creation has been worthwile, he will also consider the set of natural numbers that ever have been used by his creatures. And he will find that only a very small subset has been applied. (This Idea goes back to Borel, cp. § 80.) For every usable number we have a finite set of predecessors and an infinite set of ¡0 successors. So there is no usable natural number behind some borderline, although that border line cannot be determined yet. Is it, in principle, possible to find circumstantial evidence for the existence of the ¡0 inaccessible numbers - in order to satisfy platonists like Gödel? Or is postulating them by the axiom of infinity the only way to lay hold of them?

§ 296 The following three characteristics of the horizon are now important for our theme. Firstly, we do not understand the horizon as the boundary of the world, but as a boundary of our view. So the world continues even beyond the horizon. Secondly, the horizon is not some line drawn and fixed in the world but it moves depending on the view in question, specifically on the degree of its sharpness. The further we manage to push the horizon, the sharper the view. Thirdly, for a phenomenon situated in front of the horizon, the closer it is to the horizon, the less definite it is. [P. Vopenka: "The philosophical foundations of alternative set theory", International Journal Of General System 20,1 (1991) 115-126.]

§ 297 What would correspond more to the spirit of physics would be a mathematical theory of the integers in which numbers, when they became very large, would acquire, in some sense, a "blurred" form and would not be strictly defined members of the sequence of natural numbers as we consider it. The existing theory is, so to speak, over-accurate: adding unity changes the number, but what does the addition of one molecule to the gas in a container change for the physicist? If we agree to accept these considerations even as a remote hint of the possibility of a new type of mathematical theory, then first and foremost, in this theory one would have to give up the idea that any term of the sequence of natural numbers is obtained by the successive addition of unity - an idea which is not, of course, formulated literally in the existing theory, but which is provoked indirectly by the principle of mathematical induction. It is probable that for "very large" numbers, the addition of unity should not, in general, change them (the objection that by successively adding unity it is possible to add on any number is not quoted, by force of what has been said above). [P.K. Rashevskii: "On the dogma of the natural numbers", Russian Mathematical Surveys 28,4 (1973) 143-148]

Compare the pocket calculator with 1010 + 1 = 1010. And forget Cantor's impracticable ω + 1 > 1 + ω.

§ 298 The scheme of construction of a mathematical theory is exactly the same as that in any other natural science. First we consider some objects and make some observations in special cases. Then we try and find the limits of application of our observations, look for counter- examples which would prevent unjustified extension of our observations onto a too wide range of events [...]. I even got the impression that scholastic mathematicians (who have little knowledge of physics) believe in the principal difference of the axiomatic mathematics from modelling which is common in natural science and which always requires the subsequent control of deductions by an experiment. Not even mentioning the relative character of initial axioms, one cannot forget about the inevitability of logical mistakes in long arguments (say, in the form of a computer breakdown caused by cosmic rays or quantum oscillations). Every working mathematician knows that if one does not control oneself (best of all by examples), then after some ten pages half of all the signs in formulae will be wrong and twos will find their way from denominators into numerators. The technology of combatting such errors is the same external control by experiments or observations as in any experimental science and it should be taught from the very beginning to all juniors in schools. {{Great scepticisms appears appropriate when something simultaneously is asserted to be attainable and simultaneously is asserted to be unattainable. Experimental verification of matheological results is not possible. You are depending on the statements of logicians. By the way, according to Thomas Mann, literati are people who find writing difficult. Transferred to logic and set theory, logicians are people who find thinking difficult.}} [V.I. Arnold: "On teaching mathematics" (1997) Translated by A.V. Goryunov] http://pauli.uni-muenster.de/~munsteg/arnold.html

§ 299 There are many examples of "soritic properties" for which mathematical induction does not hold ("number of grains in a heap", "number that can be written down with pencil and paper in decimal notation", "macroscopic number", ... ), but mathematicians traditionally take no account of them in their theories, with the excuse that such properties are vague. We present here a mathematically rigorous theory in which a soritic property is put to constructive use. [Karel Hrbacek, Olivier Lessmann, Richard O'Donovan: "Analysis with ultrasmall numbers", Amer. Math. Monthly 117,9 (2010) 801-816]

§ 300 Let F(n) = {1, 2, 3, ..., n} be the n-th finite initial segment of the set of natural numbers. 100 100 Then the sequence (an) defined by an = min{ 10 , |Ù \ F(n)| } = 10 has limit 0 in matheology but limit 10100 in mathematics. So matheology broke the bands of mathematics.

§ 301 MadOverflow

Richard Dawkins' phrase "I regard Islam as one of the great evils in the world, and I fear that we have a very difficult struggle there" would be very hard to defend in Pakistan. Probably Dawkins would be deleted. That would prove him right. But he hardly would enjoy it.

I regard matheology [i.e., the belief in finished infinity and in numbers which cannot be (finitely - how else?) defined, which cannot be used as individuals in mathematics and elsewhere and which cannot be known at all] as one of the great evils in academia, and I fear that we have a very hard time to clear the brains from that confusion. Fortunately this leads only to deletion of my explanatory contributions to MathOverflow, a forum of "research-level mathematics" where quite a lot of strange things happened. Why? Well, one of the moderators has already provided the answer: "This is a field particularly prone to incompetence so severe as to make recognizing one's own incompetence impossible." [Scott Morrison, Jul 11 at 16:56] http://meta.mathoverflow.net/questions/435/the-association-bonus

In order to dismiss accusations of incompetence let me mention that I wrote three text books of mathematics, one of them available in seventh edition meanwhile and another one having acquired bestseller status with a renowned publisher. Let me mention further that I have collected more than 1000 reputation points in MatheOverflow within less than three months of activity - just for fun and of course under pseudonym because my signed contributions usually have not a long lifetime in that forum.

I will report here in sci.logic 50 paragraphs of those events which really are so strange that one could believe the reports stem from a forum called MadOverflow.

§ 302 Can family planning change the equipartition of boys and girls?

In a country in which people only want boys every family continues to have children until they have a boy. If they have a girl, they have another child. If they have a boy, they stop. What is the proportion of boys to girls in the country? http://www.businessinsider.com/answers-to-15-google-interview-questions-that-will-make-you- feel-stupid-2009-11#in-a-country-in-which-people-only-want-boys-3 Google claims that a 50/50 population will be maintained. Family planning cannot influence birth- probability.

The easiest proof is this: Imagine that boys and girls to be born come on a great conveyor belt, distributed at a 50/50-ratio. In case a woman wants to have a child she takes the next one from that conveyor belt. So it is clear that the ratio will not depend on the question whether or not the woman had already born one or more girls.

This simple fact is understood by all biologists and all other scientists which I know of - alas not by all users of MathOverflow, since it is not "research-level" mathematics. In MathOverflow things look different. Douglas Zare in a "great answer" has calculated the percentage of girls, G/(G+B) per family, and has found that the average of this ratio deviates from the correct answer and depends on the number of couples involved. http://mathoverflow.net/questions/17960/google-question-in-a-country-in-which-people-only- want-boys-closed

Instead, however, doubting his result, he writes: “So, for a large population such as a country, the official answer of 1/2 is approximately correct, although the explanation is misleading. In particular, for 10 couples, the expected percentage of girls is 47.51% contrary to what the official answer suggests.” This "great answer” is not even approximately correct (simple example: if you want to know whether the sum of numerators is equal to the sum of denominators, you cannot calculate the average of fractions) but it has been approved by far more than 100 "professional” mathematicians and logicians in the self-proclaimed elite forum MathOverflow.

Zare is strongly supported by Steven Landsburg, who even accepts bets over 5000 $ and has accused the Harvard string theorist Lubos Motl (as well as others) of “not knowing enough mathematics”. Landsburg asks in his blog: Are You Smarter Than Google? There’s a certain country where everybody wants to have a son. Therefore each couple keeps having children until they have a boy; then they stop. What fraction of the population is female? http://www.thebigquestions.com/2010/12/21/are-you-smarter-than-google/

Landsburg explains the terms of the bet: The best grad studens of the top 10 elite universities should write simulations to get results with reliable statistics. Obviously he has not recognized that every random binary sequence will do which can be easily obtained just by throwing a coin.

A mathematician calling himself Monty has figured that out: Consider a very long random sequence of bits. If only partial sequences ending with a 1 (put 1 for boy, 0 for girl) are taken, like in this sequence: 01|01|001|1|001|0001|1|01|1|01|01|1|1| then this sequence would simulate many countries with only one couple each {{as well as one country with many couples and every other mixture}}. Every sequence stops when a 1 is returned. The average of the ratios G/(G+B) is in fact as caculated by Douglas Zare about 31 %.

Alas, giving this answer does not mean to be smarter than Google! It answers a completely different question. The question asks for the expected ratio of all girls (or all boys) to all girls and boys, and not for the average of these ratios per family. Zare could have learned the difference from Monty, who was about to ask why Zare had used G/(G+B) per family and not B/G per family. In the latter case the average would have been infinity, because those families who stopped after having a boy as the first child contribute B/G = ¶. Douglas Zare and Steven Landsburg and their over one hundred supporters then would have learned that they are far from being smarter than Google. But Monty's answer had already been deleted before he could provide his explanation.

§ 303 The digits of π

I know it's likely that, given a finite sequence of digits, one can find that sequence in the digits of π, but is there a proof that this is possible for all finite sequences? Or is it just very probable? [sep332, March 2010] http://mathoverflow.net/questions/18375/is-there-any-finitely-long-sequence-of-digits-which-is- not-found-in-the-digits-of/132035#132035

The answer is no: It is neither provable nor even probable because it is provably impossible to find out. The set of n-digit-strings contains at least one string that has Solomonoff- or Kolmogorov- complexity n. See http://people.cs.uchicago.edu/~fortnow/papers/kaikoura.pdf The complexity of a program is limited by the ressources available in the memory of the computer. The accessible universe contains an upper limit of less than 10100 bit. Therefore it is even impossible, and will remain so forever, to define "every finite string of bits" and it would be unscientific speculation to argue whether all finite strings are in the decimal representation of π. It can even be excluded that a natural number with a complexity of 10100 bit can be defined and applied, let alone the corresponding digit of the decimal expansion of π. [AGreen, May 2013]

This answer of course is not what "research-level-professionals" like to enjoy. It would hurt their belief in countable and uncountable sets if they realized that they have not even the chance to use one infinite set in mathematics, i.e., such that every element has its own name and identity. They confuse the possibility to calculate the limit of an infinite sequence with the impossibility to calculate infinitely many terms of that sequence. Therefore the answer got a lot of downvotes and finally was deleted.

And Douglas Zare, the genius who had proven that family planning can change the ratio of girls in a country (see § 302), again shows his superiority by stating: en.wikipedia.org/wiki/Champernowne_constant can be proven to be normal. I find it very odd that 3 people have voted up a supposed answer with a huge error like this. [Douglas Zare, May 2013]

Obviously Zare does not or cannot distinguish between the two propositions: "There exist normal numbers like the Champernowne constant" and "all irrational numbers including π are normal".

§ 304 Everything is consistent in any case

"It is impossible in principle to well-order the reals in a definable manner." To be more precise, the belief I am talking about is the belief that well-orderings of the reals are provably chaotic in some sense and certainly not definable. For example, the belief would be that we can prove in ZFC that no well-ordering of the reals arises in the (that is, definable in the real field, using a definition quantifying over reals and integers). This belief is relatively common, but false, if the axioms of set theory are themselves consistent {{and that can be taken for granted since}} the idea nevertheless has a truth at its core, which is that although it is consistent that there is a definable well-ordering of the reals (or the universe), it is also consistent that there is no such definable well-ordering. {{That's what I like with ZFC. There are no inconsistencies!}} [Joel David Hamkins (2010), giving in MathOverflow an example of common false beliefs in mathematics] http://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics- closed

§ 305 Interpretation of Limits - Limits of Interpretation?

Consider this sequence

21. 2.1 432.1 43.21 6543.21 654.321 87654.321 8765.4321 1098765.4321 109876.54321 1211109876.54321 121110987.654321 .. .

Its limit in analysis is infinity.

However, we can also interpret this sequence as a supertask of set theory. We put in always two natural numbers and remove always one natural number. The input is added on the left-hand side, the the output passes the point for the right-hand side. Then “after having completed” the supertask, all numbers have been removed from the left-hand side. Therefore the limit in set theory is less than 1.

After appearing for a short time in MathOverflow and MathStackExchange, this observation got deleted. But there has been an unmasking comment by the user Michael Greinecker: "There is no contradiction. Just a somewhat surprising result. And there is no a apriori reason why one should be able to plug in cardinal numbers in arithmetic formulas for real numbers and get a sensible result."

Well, we start with only positive integers, i.e., natural numbers or positive finite cardinal numbers. If however, we decide to interpret their concatenation as a real number, then we get a different limit. In mathematics limits do not depend on an unspecified “interpretation”. In matheology, however, facts (or what is taken as facts) are different.

§ 306 The deal with the devil

In the paper http://arxiv.org/PS_cache/math/pdf/0212/0212047v1.pdf J.D. Hamkins describes a deal with the devil after which you have lost all your money.

Thus, on the first transaction he accepts from you bill number 1, and pays you with bills numbered 2 and 4. On the next transaction he buys from you bill number 2 (which he had just paid you) and gives you bills numbered 6 and 8. Next, he buys bill number 3 from you with bills 10 and 12, and so on. When all the exchanges are completed, what do you discover? You have no money left at all! The reason is that at the nth exchange, the Devil took from you bill number n, and never subsequently returned it to you.

I think I can contribute an idea that saves you at least one bill. For that sake simply require that you never hand a bill to the devil unless you are in possession of at least another bill. This condition, laid down by an additional clause in the contract, must never be violated and remains valid in eternity. Or put it the other way round: In case you would go bancrupt, simply refuse to hand him the last bill you have. (There is no last natural number and of course no "time ω" where everything including the Cantor-diagonal could happen, but before you are in possession of zero bills, you must have been in possession of at least one - and you never lose bills by handing out more than a single one.) This will prevent your total bancrupt, won't it? It will not, however, prevent the total bancrupt of some Cantor-cranks here around. They cannot defend their ridiculous position that devil breaks contract. He never does. Therefore they must delete this text as quickly as possible in order to keep the respect of their poor students. [Devil, 9 July 2013]

Dear Devil, I am voting to close since this is not a real question. Your statement is: if you change the rules of the game, then you also change the outcome. I don't think anyone disagrees with that principle, it's just that there is no research level math in your "question". [Vidit Nanda, 9 July 2013]

Sorry, I don't change the rules of the game. This condition is always in every super task implicitly realised. No natural number is without a successor. Therefore it is nonsense to talk about all numbers or to enumerate all rationals. [Devil, 9 July 2013]

It seems that most set theorists do not realize that there is no chance to issue a last natural number. The reason is that there is no last one. But that also implies that you never can issue all natural numbers. Does this fact restrict mathematics? Some seem to think so:

And also, apparently, we cannot walk from here to there. en.wikipedia.org/wiki/Zeno's_paradoxes [Joel David Hamkins, 9 July 2013]

If this supertask has to do anything with the arrival of Zeno’s arrow, i.e., with reality, then the result of the deal (like anything in reality) must not depend on how the bills are enumerated. Then the dealer with the devil can withhold an arbitrary amount of numbers and will acquire so an infinity of wealth.

No, reality is not saved by matheology!

§ 307 Requires set theory accepting the result of a super task?

Littlewood's super-task has been vividly illustrated in this question: http://mathoverflow.net/questions/7063/a-problem-of-an-infinite-number-of-balls-and-an-urn But not all mathematicians and very few scientists and philosophers accept that "finally the urn is empty". One user wrote:

I am a mathematician and I don't accept "0 balls" as "the" answer (because of implicit continuity assumptions). There is ample evidence at this page that I am not alone in that. [Victor Protsak 26 May 2010]

That is certainly true. However, mathematicians who deny the null-result but accept the notion of countability are not aware of the fact that every enumeration of a countable set is a super task. Counting the positive rational numbers, for instance, yields the following super task: In the nth step fill in all rationals between n-1 and n and, if not yet residing in the urn, also the rational number qn to be enumerated by the natural number n. Take off the rational number qn. Go to step n+1. Here we have in every step infinitely many numbers going in and only one going out. Fraenkel has illustrated this fact by the well known story of Tristram Shandy.

Therefore all mathematicians who don't accept the null-result should not accept countability as a sensible notion, let alone uncountability.

Those however, who accept the null-result, should try, for a moment, to imagine the possibility to delay the enumeration a bit by detaining always a natural number until another one is available http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT#394,42,Folie 42. Applied to the deal with the devil this method prevents the total bankrupt of the player, applied to the enumeration of the positive rational numbers, this method prevents their complete enumeration.

The most natural property of the natural numbers is that it is never possible to use a last one such that none remains unused. And here "never" means never, and not "in some infinity". But if things can happen, then: - The urn: For every step n the contents increases. "At infinity" the urn is empty. - The Cantor-list: Up to every line n, the diagonal is not contained. "At infinity" the diagonal is there.

No axioms available that distinguish between these two cases.

My rational question: Is there a way (other than pure belief in the impossible) to circumvent the fact that the natural numbers never can be exhausted?

This question acquired 7 downvotes in MathOverflow and was deleted soon. Matheologians must be very afraid of rational thought.

§ 308 Others than all?

On 3 July 2013 Rainbow asked in MathOverflow: Every path in a complete infinite tree represents a real number of the unit interval between 0 and 1. (Some rational numbers have two representations.) Assume that you have a can of red paint and that you can colour an infinite path of the tree with one can. Assume further that you get another can of red paint for every node that you are colouring for the first time. Then you will first accumulate an infinity of cans of red paint, but you will nevertheless not be able to colour all paths, since there are uncountably many paths in the tree (and you can win only countably many cans of red paint). So there remain uncountably many paths uncoloured in the first run. Start with another colour, say green. Also with green paint you will not finish. How many different colours will be required?

Andreas Blass replied: "You seem to be ignoring the fact that, after you have colored a countable family of pathes, say P0, P1, …, Pn, …, there may be other paths Q that are not on this countable list but have, nevertheless, had all their nodes and edges colored."

I cannot know what Rainbow thinks of this answer. I, for my person, think, that “other” paths must be different from the paths of the family of coloured paths in order to be “other”. So there must be at least one node differing from the nodes of a coloured path. But if we leave a coloured path, we can only do so by using a node of another path that is also a coloured path (since every path has been coloured). That means we never leave and always remain on a coloured path. It seems to me that the “fact” Andreas Blass ist talking about refers to a . But in the tree we have not a list. Nevertheless there are provably only countable many paths. And all are coloured red.

This may be the reason why Rainbow’s answer has been deleted soon. Nobody would know about the dyslogic applied by Andreas Blass, unless I had, by chance, copied all this stuff.

§ 309 Two identical sequences with very different limits

When constructing the Binary Tree node by node, we need ¡0 steps. Compare for instance http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT. In the limit the complete Binary Tree contains 2¡0 paths representing, among others, irrational numbers. Therefore the cardinality of the set of irrational numbers constructed up to step n can be understood as a sequence f(n) = 0 for every n with limit 2¡0. My question: Has this been observed and discussed in literature already?

And may I add a second question: If we construct, line by line, a Cantor-list containing all rational numbers, then the cardinality of the set of irrational numbers contained in that list up to line n is a sequence g(n) = 0 for every n with limit 0.

Is there any result in research level mathematics why two zero-sequences show such blatantly different behaviour?

This question got 7 downvotes in MathOverflow but also 3 upvotes. Are there in fact real mathematicians remaining in this Gomorrah of matheology?

§ 310 How to distinguish between the complete and the incomplete infinite Binary Tree?

How can we distinguish between that infinite Binary Tree that contains only all finite initial segments of the infinite paths and that complete infinite Binary Tree that in addition also contains all infinite paths? Let k denote the kth level of the Binary Tree. The set of all nodes of the Binary Tree defined by the union of all finite initial segments of the sequence of levels »0

This question got 7 downvotes in MathOverflow but also 2 upvotes. Are there in fact real mathematicians remaining in this Gomorrah of matheology?

I note only three further comments of mine: I cannot understand why some cranks here rate my question negative? I am sure, they have never pondered about it and cannot answer it. Fortunately some mathematicians seem to have a larger horizon than their standpoint. ZF minus infinity is the smaller tree here. But if the existence of the larger tree is claimed and the Axiom of Infinity is assumed, then it must be in accordance with mathematics. Otherwise it is as invalid as the axiom stating the existence of 10 different natural numbers with sum 10. And I am asking whether someone can support the Axiom of Infinity as sensible. A level ω is certainly not the solution - at least as far as mathematics is concerned. Here is a simpler explanation: Try to distinguish the set of all terminating fractions and the set of all real numbers by digits. And if you cannot, try to explain why Cantor's diagonal argument is said to apply to actually infinite decimal representations only. Perhaps even try to understand, why I claim that everything in Cantor's list happens exclusively inside of finite initial segments, such that, in effect, Cantor proves the uncountability of a countable set. § 311 How many is ω? Is it larger than ω?

If we count the digits of a real number, we get the result ω. So for every rational approximation pn we get a digit where it differs from π. On the other hand there are also ω rational approximations which, up to n, do not differ from π. If we consider this in the finite case, say 3, then 123 cannot differ from all of its three finite approximations 1, 12, and 123. Any suggestions how to remedy that gap in case of ω? Edit: How can a real number be distinguished from all its rational approximations? It can't. Not by digits. But all rational approximations form a countable set. So the real numbers cannot be defined other than by finite definitions, which form a countable set (in every speakable language that can be used in mathematical discourse). I am afraid the answer to this question is a bit above the "research level" prevailing here. But who knows?

Questions that cannot be answered without confessing that transfinity is inconsistent are usually deleted immediately - like the present one.

§ 312 The odd even number ω

The following question http://mathoverflow.net/questions/21457/are-there-an-odd-number-of-even-numbers-closed asked on 15 April 2010 at 13:23 by Alex Andronov got a surprising comment:

Yes, the first infinite ordinal ω is even, but other infinite ordinals, such as ω + 1 or ωω + 5 are odd ordinals. The cardinality of the set of even natural numbers is ω which is even. In fact, under AC, all infinite cardinals are even. [Joel David Hamkins, 15 April 2010]

Unfortunately, meanwhile not even rudiments of this question are available any longer.

§ 313 A remarkable result of matheology

If we colour every node of some path of the Binary Tree, say the path 0.000…, then we have coloured this path: 0. 0 0 0 ∂

But if we colour in addition to every node all its predecessors, then we have not coloured that path: 0. 0.0 0.00 0.000 ∂

This remarkable observation concerns also the definition of paths: If we define every node of some path of the Binary Tree, say the path 0.0101010… = 1/3, then we have defined this path. But if we define in addition to every node all its predecessors, then we have not defined this path 1/3.

The reason of the latter is that none of the finite initial segments of the binary representation of 1/3 is 1/3. The reason of the former is, that, although none of the bits finishes the binary representation of 1/3, we nevertheless believe, inspired by Cantor, that a binary representation of 1/3 exists and can be finished. (Of course the “we” in this text has been purest sarcasm.)

Perhaps because of such observations some matheologians are really afraid: “Wolfgang Mückenheim is probably one of the most dangerous cranks out there. He has a professorship at the University {{of Applied Sciences}} of Augsburg, Germany, where he is teaching physics and mathematics!! Currently {{having started in 2003}}, he is teaching a lecture called "History of the Infinite". This man does real damage.” [Michael Greinecker, 1 May 2012] http://meta.mathoverflow.net/discussion/1353/nominalist-foundations-of-mathematics/

Thank you, Michael. I hope so because exactly that's my intention. Damaging the nonsense that you Cantor-cranks call mathematics but that is merely a perversion of mathematics, i.e., matheology. (For a definition, see § 001.)

§ 314 Undefinable “reals”?

In order to define uncountably many “real” numbers, infinitely many bits or other letters of a countable alphabet are required, but nobody can send or receive infinitely many letters. So you need an uncountable alphabet. Since undefined letters are not suitable to carry information, the letters of an uncountable alphabet have to be defined first, for sender and receiver. In order to define uncountably many letters such that they can be used in mathematical discourse, you need uncountably many sequences of bits. It is no secret that most of these sequences must be infinite. But infinite sequences cannot be used (written, submitted, received) other than by their finite names. So the infinite sequences must be defined by finite names constructed from a countable alphabet . Alas, there are only countably many such finite names available. Therefore, to commit a gross understatement, not all letters of an uncountable alphabet can have different definitions. Every single definition has to define uncountably many different letters of the uncountable alphabet. But letters with identical definitions are scarcely different, and to guess which of uncountably many letters is meant by a certain definition, is a task that is impossible to accomplish in mathematics. That is only possible in matheology – because of important non- written special information processed and submitted by means that remain unknown outside of the inner circle of matheologians.

§ 315 Is subcountability as powerful as uncountability in confusing mathematicians?

Andreas Blass stated in MathOverflow (meta) that "eventually, most mathematicians came to accept that definability should not be required, partly because the axiom of choice leads to nice results, but mostly because of the difficulties that arise when one tries to make notion of definability precise." (meanwhile deleted)

That is a real surprise to me. Which mathematicians accepted that and when? Have they been summoned to a public meeting with voting about abolishing definitions like the astronomers when Pluto was degraded?

Wouldn't a set with undefined elements contradict the Axiom of Extensionality: “If every element of X is an element of Y and every element of Y is an element of X, then X = Y.” How could that be decided for undefined elements?

Wouldn't a set with undefined elements contradict the Axiom of Choice? How could we choose something that cannot be chosen, in spite of the axiom, because we cannot say or express in any other way what we intend to choose? Blind choice - the foundation of matheology? It resembles the attempt of brainless thinking.

But my actual question is this: I have heard of another solution. The set of finite definitions is countable. That cannot be explained away, can it? But not every finite definition has a meaning. In fact, if we refrain from using common sense, we cannot even define definability, let alone the set of meaningful definitions. Therefore this set is not countable but subcountable - and if we identify subcountability with uncountability, we have won and can continue to enjoy the nice results of the axiom of choice.

Of course this question has been deleted soon.

§ 316 Does undefinable definability save matheology?

On 13 June 2013 user albino (meanwhile deleted) asked in MathOverflow about set theory without the axiom of power set. http://mathoverflow.net/questions/133597/what-would-remain-of-current-mathematics-without- axiom-of-power-set I will report also the interesting subsequent discussion about definability.

The power set of every infinite set is uncountable. An infinite set (as an element of the power set) cannot be defined by writing the infinite sequence of its elements but only by a finite formula. By lexical ordering of finite formulas we see that the set of finite formulas is countable. So it is impossible to define all elements of the uncountable power set. [albino]

You say that the powerset of an infinite set is questionable because it must have some undefinable elements. You are presuming that the subsets of a set must all be distinguishable by you, or some entity whose only access to powersets is through formal language. But why is such an assumption warranted? What makes you think that a thing does not exist unless you can define it? Is existence a personal belief? [Andrej Bauer]

I think that elements of a set must be distinguishable, as Cantor has put it. [albino]

{{I would say that the existence of personal beliefs is a personal belief. And I think that undefinable elements cannot be applied. How would you apply an element that you cannot define? What are elements good for that cannot be applied in mathematics? Matheology. But unfortunately I came to late to take part in this discussion}}

Also, you are confusing "distinguishable" with "distinguishable by a formula". [Andrej Bauer]

{{He said so but refused to explain the difference. By the way, every distinction in mathematics occurs by a finite formula.}}

I am afraid I have not got the meaning of your sentence you are confusing "distinguishable" with "distinguishable by a formula". If you cannot get hold of a notion other than by a finite formula, how would you distinguish two of them without a finite formula? [albino]

You have focused on this definability issue, but that's just not crucial. I heartily recommend that you read Joel Hamkin's post that he linked to in the comments to the question. He explained very well why definability is a deceptive notion. [Andrej Bauer]

{{I think that a person who distinguishes between "distinguishable" and "distinguishable by a formula" but refuses to explain this, is a deceptive person.}}

… Lastly, concerning your remarks about definability, I refer you as I mentioned in the comments to an answer I wrote to a similar proposal, which I believe show that naive treatment of the concept of definability is ultimately flawed. [Joel David Hamkins]

Logicians, I have learned, take some premises and obtain some conclusions. But they do not judge about truth or practical things. In my opinion "definability" is a practical notion. If I define something and others are able to understand what I have defined, than that something is definable (otherwise it may be undefinable or I am not good enough in defining). But with respect to numbers things are easy. If I say π or e or 1/4, then these numbers are defined. And it is true, in my opinion, that not more than countably many numbers can be defined by finite strings of bits. [albino]

Albino, I'm not sure to which logicians you are referring with your first comment. Meanwhile, yes, your remarks on definability are the usual naive position on definability. If you ever find yourself inclined to mount a serious analysis of definability, however, then I would suggest that you talk more with logicians. {{Really? Why are modern logicians despised like parias by all kinds of scientists?}} In particular, I would point you toward the initial part of my paper on pointwise definable models of set theory (de.arxiv.org/abs/1105.4597), where we deal with the "Math Tea argument", which is essentially the argument you are advancing. [Joel David Hamkins]

I am sorry, but since Zermelo's “Beweis, daß jede Menge wohlgeordnet werden kann”, we know that we have to distinguish between logical proofs and real proofs. If you had proven that all numbers can be expressed with three digits, I would not believe you. And your proof is rather similar. So I know it is not a real proof. Nevertheless I read your paper up to Theorem 4. It reminds me of Zermelo's “If AC, then well-ordering is possible”. But I am not interested in that kind of logical conclusions but only whether I can do it. [albino]

Oh, I'm very sorry to hear that you aren't interested in logical proof or logical conclusions. I'll leave you alone, then, to undertake your own kind of proof activity. [Joel David Hamkins]

I do accept logical proof! I accept the logical proof that it is impossible to define more than countably many objects including all numbers. This stands as solid as the proof that with three digits you cannot define more than 999 natural numbers. I even accept non-constructive proofs like Zermelo's, but not as deciding whether something can be constructed - as was Zermelo original intention. (Compare Fraenkel who said that hitherto nobody could well-order the reals.) To be short: If your proof is correct, then you have found a contradiction. [albino]

{{I don't think it is a good idea to ask logicians what can be done in reality. Zermelo was the first to make a fool out of himself, when insisting and "proving" by insisting that every set can be well- ordered, Hamkins will not be the last one. To put an axiom may be a good idea in order to find out what can be thought - but not what can be done.}}

§ 317 Violation of inclusion monotony in infinite applications?

Ex Oriente Lux asked in MathOverflow: Consider the sequence of ordered subsets of natural numbers, written below each other in form of a triangular matrix:

(1) (2, 1) (3, 2, 1) ...

By inclusion monotony we see that every number that is contained in the matrix is in one of its lines together with all other numbers contained in this matrix. Each vertical row contains ¡0 natural numbers. Since each vertical row is a subset of the matrix, we find that ¡0 numbers are contained in the matrix. From every horizontal line we see that only a finite set of numbers is in that matrix, since there is no line with ¡0 natural numbers. If there are ¡0 natural numbers in the matrix, they cannot be in one and the same line. This fact violates inclusion monotony. Is it worthwile to accept ¡0 as a number larger than every natural number when it must be payed for by the disadvantage that inclusion monotony can no longer be trusted?

This very simple question has been deleted soon, as usual if matheologians cannot answer without unveiling a contradiction between mathematics and the idea of competed infinity. But the result is without the least doubt: Either there are not "all" natural numbers, i.e., a complete set Ù such that no further natural can be added, or Ù is contained in all lines of the list without being in one line of the list although there are "all" lines such that none can be added.

This same effect appears in the Binary Tree, when asking for the tree containing only all finite initial segments of the infinite paths. Some hold that such a tree contains automatically all infinite paths nevertheless. But then the list

0.1 0.11 0.111 ... must also contain the number 1/9 (although it is in no single line, it must be there somehow). Or they try to distinguish between the tree with and without "nodes at level ω". Compare § 198.

§ 318 Does the contents of a set depend on the notation?

Ex Oriente Lux asked in MathOverflow: Does the contents of a set depend on the notation of its elements? The real numbers of the unit interval [0, 1] can be represented as paths (in a Binary Tree) or as sequences of bits (in the usual manner).

Examples 1 = 0.111... = 0-1-1-1-..., or 1/3 = 0.010101... = 0-0-1-0-1-0-1-...

The set of all rational numbers of the unit interval can be enumerated. This sequence does not contain any irrational numbers. But if the same set is written in the form of a Binary Tree, then all irrational numbers sneak in, since it is impossible to construct the Binary Tree leaving out the irrational numbers.

My questions: 1) How do all irrational numbers of the unit interval sneak into the Binary Tree that is constructed only by paths representing rational numbers? Is there an intuitive picture saying that at the end or slightly before or after the end uncountably many irrationals enter?

And 2) If we define the function f(rational x) = 1 and f(irrational x) = 0, are then the irrationals also automatically in the set of rationals with f(x) = 1, like they are always in the Binary Tree, such that the function f is 1 everywhere? Or are they not in this set like in a list of only rationals?

And 3) why is this as it is?

This great question quickly earned 6 downvotes and was deleted after three hours. So it must be a very dangerous one.

§ 319 Is it possible to find such permutation of natural numbers that it cannot be a limit of finite permutations? This question has been asked by kakaz on 9 March 2010 in MathOverflow. http://mathoverflow.net/questions/17653/infinite-permutations

The main answer, provided by J.D. Hamkins on the same day, said no. It got 6 upvotes, only one less than the question itself.

Another answer, supplied by Luitzen Egbertus Jan on 3 July 2013 got no upvotes but is correct (well he had much more time to think it over). He said yes:

I think the limit of infinitely many permutations of the set of natural numbers Ù can be non existent. Consider an enumeration of the rationals –: q1, q2, q3, q4, q5, q6, q7, ..., for instance the classical enumeration given by Cantor. Apply operation A: order the rationals pairwise such that q2k+1 and q2k+2 belong to a pair (q1, q2), (q3, q4), (q5, q6), (q7, ... and order the rationals of the first n parentheses by their magnitude such that the smaller number comes first. Enumerate them in the new order. Apply operation B: order the rationals pairwise such that q2k and q2k+1 belong to a pair q1, (q2, q3), (q4, q5), (q6, q7), ..., and order the rationals of the first n parentheses by their magnitude such that the smaller number comes first. Enumerate them in the new order. Repeat operations A and B alternatingly for infinitely incresing but always finite numbers n of parentheses. The limit would be a well-ordering of all rational numbers by magnitude.

What is the meaning of “all”? Like in Cantor’s enumeration we can continue until we have included any desired rational number that was fixed in advance. So if we agree that Cantor was able to enumerate all rational numbers by showing that he can enumerate every desired finite number of rationals such that the relative position of the enumerated rationals will remain the same in every following step, we can be sure to have well-ordered by magnitude all rational numbers because we can well-order every desired finite number of rationals such that the relative positions of these n rationals will remain the same in every following step (and only after fixing that we add another rational).

Since a well-ordering by magnitude is impossible, the corresponding limit of the natural numbers does not exist.

§ 320 Sébastien Palcoux complained in MathOverflow: I had problems with a question (closed in 9 min, 10 down votes and finally deleted) about mathematicians, neurosis and all common stereotypes among non-mathematicians on that. I warn you that I'm not talking about severe neurosis as schizophrenia (John Nash ...), it's not at all my point. My point is about mild neurosis, allegedly widespread (by non-mathematicians) among mathematicians.

§ 321 Prime Mister asked in MathOverflow: Enumerate all rational numbers to construct a Cantor-list. Replace the diagonal digits ann by dn in the usual way to obtain the anti-diagonal d. Beyond the n-th line there are f(n) rational numbers the first n digits of which are the same as the first n digits d1, d2, d3, ..., dn of the anti-diagonal. f(n) is infinite for every n. So we can safely say that we can find n duplicates of the first n digits d1, d2, d3, ..., dn of the anti-diagonal. Define for every n the function g(n) = 1/n. In analysis g(n) has the limit 0. So in the limit there are infinitely many replica of the diagonal in the list.

This question soon got deleted. The general tenor is: Although every digit of d and all its predecessors are in a line (in fact in infinitely many lines) of the list, d is not in the list. But can d be defined by more digits than all numbers in the list can contribute?

Now consider a list of all finite initial segments of Ù

1 1, 2 1, 2, 3 ...

Applying the same reasoning as above, we can say that Ù is not in the list. But we know that all natural numbers are in the list. This means all natural numbers are there but not in one line. So they must be distributed over more than one line. That means there must be more lines, at least two lines, that together contain more natural numbers than any of them. Contradiction by the fact of inclusion monotony of the list.

A similar contradiction results from this list:

0.1 0.11 0.111 ...

1/9 is the limit of the listed sequences, but it is not in the list. So if the existence of a decimal representation of 1/9 is asserted, then it must be distributed over more than one line of the list. Contradiction by the fact of inclusion monotony of the indices of the sequence of indices.

Result: All that exists of the Cantor-diagonal d, of Ù, and of 1/9's decimal representation, is in one line of the list. But all that is not all.

§ 322 Breaking circularity

Some days ago, I posted a question about models of arithmetic and incompleteness. I then made a mixture of too many scattered ideas. Thinking again about such matters, I realize that what really annoyed me was the assertion by Ken Kunen that the circularity in the informal definition of natural number (what one gets starting from 0 by iterating the successor operation a finite number of times) is broken “by formalizing the properties of the order relation on omega” (page 23 of his “The Foundations of Mathematics”). What does actually “breaking the circularity” mean? Is there a precise model theoretic statement that expresses this meaning? And what about proving that statement? Is that possible? [Marc Alcobé García, 14 Jun 2010]

Arithmos answered in MathOverflow in July 2013:

It is impossible to break the circularity immanent in every definition, unless you define some primitive notions as non-circular. Nevertheless they are. Every single word in "there is a set s, such that the empty set is an element of s ..." is undefined or defined by other words that are undefined. This deficiency is not remedied but at most veiled by writing the axiom of infinity or calling something ω.

Everybody has to begin with some words that he has learned when his mother taught him his mother-tongue. Therefore it is no surprise that there is much ado about definability, which, again, is undefinable, at least in first order logic.

Having recognized this, it is irrelevant what primitive notions exactly you start with. Before defining the natural numbers, students should know how to count and, therefore, how to add the unit. Then the following three axioms can be used:

- 1 is a natural number. - If n is a natural number, then n + 1 is a natural number. - Everything else is not a natural number. (But if you know something else, you know that anyway.)

To my knowledge, these axioms are the only axioms that define the natural numbers and only them, namely their sequential character, their inductive property, and their equidistance.

Why use the notion "successor"? Is "successor" more primitive than "addition of the unit"? No. Knowing what a successor is, is as easy and as difficult as knowing what +1 means - in real life as well as in formal definitions.

But what is more important, the successor-definition includes infinitely many sequences isomorphic to the sequence of the natural numbers like:

1, 1/2, 1/3, ... 1, 11, 111, ... 1, 22, 333, ... 1, 1a, 1aa, ... {}, {{ }}, {{{ }}}, ... and even such not isomorphic to the sequence of natural numbers:

1, 11, 111, ... 0 00, 000, 000 , ... 1, square, cube, hypercube, ... 1, Fermat, Newton, Leibniz, ...

Andreas Blass commented: You correctly note that the "successor" definition needs additional information (like the that say the successor function is one-to-one and that 1 is not a successor) in order to excluse unwanted models. So does the "add a unit" definition; it might get this additional information as a special case of assumptions about the general notion of addition rather than in the form of Peano axioms, but, one way or another, the information must be included in the axiomatic base. Calling something "add a unit" rather than "successor" doesn't make a substantive difference. You may certainly believe that "add a unit" is psychologically or epistemologically a more primitive notion than "successor", but it seems to me at least as reasonable to hold the opposite opinion. I suspect that the reason "successor" is often (not always) preferred as a primitive notion is that "add a unit" looks like a special case of a general notion of addition, so authors would feel obligated to explain that only x Ø x + 1, not x, y Ø x + y is taken as primitive.

Arithmos replied: You incorrectly note that "add a unit" rather than "successor" doesn't make a substantive difference. The substantial differences are given in my examples.

§ 323 Andreas Blass, in a comment wrote on 1 May 2012 http://meta.mathoverflow.net/discussion/1353/nominalist-foundations-of-mathematics/ There have been a couple of questions recently, from someone with a very long username abbreviated to user34, attempting to promote his philosophy of mathematics. I'd paraphrase his attitude as "although I can't say precisely what I mean by definability, every sane person must agree with me that mathematics is only about definable things." {{(Paraphrasing is necessary (but not always reliable) if the text has not been fully understood.) Definability, like set, need not be defined, some even say: cannot be defined. Definitions have to be given. Mathematics is possible without sets but not without definitions. Accepting undefinable numbers cannot be excused by refuting to recognize what definability is.}} Not surprisingly, his questions have been closed and deleted; he's probably lucky that MO's software doesn't provide buttons for "tar and feather" or "draw and quarter." {{Yes, some centuries ago this would have been the method of choice.}}

§ 324 In MathOverflow Steven Landburg asked: Consider a country with n families, each of which continues having children until they have a boy and then stop. In the end, there are G girls and B = n boys. Douglas Zare's highly upvoted answer to this question computes the expected fraction of girls in the population and explains why we shouldn't expect it to equal 1/2. {{This "explanation" is grossly mistaken. Of course 1/2 is the correct answer, cp. § 302.}} My current question concerns a different statistic, namely the probability that there are more boys than girls (after all families have finished reproducing). This probability turns out to be exactly 1/2, and I'm looking for an intuitive explanation of why. http://mathoverflow.net/questions/132297/boys-and-girls-revisited

There is a very simple explanation for the lacking intuition, namely: intuition is not lacking but the expectation E[G/(G+B)] of G/(G+B) is absolutely uncorrelated to the expected answer. This would have become obvious if the original formulation (the ratio of boys to girls) had been taken literally by calculating, instead of E[G/(G+B)], the expectation of B/G which is E[B/G] = ¶. From this result certainly nobody would have concluded that the official answer is false. [Hilbert7Problem]

... it appears to be the exact opposite of the truth. If the question is "What is B/G?", and if the official answer is "1/2" {{for B/G it is 1}}, and if the correct answer is "a random variable with expected value ¶" , then recognizing the correct answer would lead not nobody, but everybody, to conclude that the official answer is false. [Steven Landsburg]

No, the expactation of fractions E(B/G) is not the fraction of expectations B/G = E(B)/E(G)! The expectation of the fractions E(B/G) is infinite, since some families have one boy as the first child and no girl. They contribute 1/0. But this does not influence "the expected fraction of girls in the population".

I can't believe that nobody in the self proclaimed "elite forum" MathOverflow opposes to Landsburg's utterings. But I find this is an extremely instructive parallel to Cantorism. Nothing could show better than this detail what an Overflow of Madness triumphs in Matheology.

§ 325 Importance of priority in mathematics?

The present question {{asked in MathOverflow by Conaeus Traglodythos on 12 July 2013}} does not ask for priority disputes, like the famous battle between Newton and Leibniz, who was the first with some invention. This question asks whether the course of mathematics has been influenced by accidental results of the historical development. As a famous example take Cantor's diagonal argument. Imagine the case that someone had discovered prior to Cantor's 1892 publication that the Binary Tree (representing all real numbers of the unit interval by infinite paths) contains only a countable set of paths that can be distinguished by nodes. The Binary Tree can be coloured or constructed completely with countably many infinite paths. Would anybody have given a dime for Cantor's argument? (This is my question.)

Alas hypothetical questions are not suitable in MathOverflow. It quickly collected five down votes and then was deleted. However, such questions remain visisble for users with 10000 reputation points. Their belief in matheology is assumed to be strong enough to read also heretical texts - like Catholic cardinals were allowed to read documents of the Vatican's safety cabinet box, like leading Nazis were allowed to enjoy Jewish music, literature and paintings, like indoctrinating fanatics always hide the truth only from the commonality (and try to punish whistle-blowers by all means).

§ 326 An unaccepted method of enumerating the positive rational numbers

On 21 July 2013 Conaeus Traglodythos asked about "A new method enumerating the positive rational numbers" in MathOverflow: It is well known that Cantor was the first one to succeed in enumerating all positive rational numbers. He obtained the sequence, the finite initial segments of which are repeated here - not because they are unknown, but because I wish to apply a new method to them: 1/1 1/1, 1/2 1/1, 1/2, 2/1 1/1, 1/2, 2/1, 1/3 and so on. This sequence will never end and for every natural number we will have accomplished much less than 10-1000000000000000000000000000000000000 of the complete task, but, since every step is well-defined and absolutely fixed, we conclude from the fact that the enumeration holds up to every rational number that the enumeration holds for all rational numbers. My idea is to apply the same method, but, in addition, always to put the finite initial segments in proper order by size (which is no problem as long as they are finite, i.e., as long as we are enumerating with finite natural numbers only - and other naturals are not known). This new method will change the sequence as follows 1/1 1/2, 1/1 1/2, 1/1, 2/1 1/3, 1/2, 1/1, 2/1 and so on. This sequence will never end, but, since also here every step is well-defined and absolutely fixed, we can conclude from the fact that enumeration and ordering hold up to every rational number that they hold for all rational numbers too. My question: Why don't we accept the second method, or, alternatively, why do we accept the first one?

Before this question got deleted, there was an interesting comment by Todd Trimble: "Ugh. When will it stop?"

Even ConTra's answer could be written and published: "Both will stop at the same instant or never." But nobody except Todd Trimble himself can know whether he could read the answer and possibly learn from it.

§ 327 On 8 April 2010 J. H. S. asked a question regarding a claim of V. I. Arnold: In his "Huygens and Barrow, Newton and Hooke", Arnold mentions a notorious teaser that, in his opinion, modern mathematicians are not capable of solving quickly. Calculate limxØ0 (sin(tanx) - tan(sinx))/(arcsin(arctanx) - arctan(arcsinx))

The answers given in http://mathoverflow.net/questions/20696/a-question-regarding-a-claim-of-v-i- arnold/132258#132258 including the accepted one (with 31 upvotes) show that professional research mathematicians in fact cannot solve such problem other than in a very ponderous way.

My answer was shorter: "I think Arnold alludes to the idea that is often used in physics lessons, for instance when treating mechanical oscillators. For small x we can put x = sinx = tanx = sintanx = arcsinarctanx etc. This yields the limit 1 immediately." [WM, 29 May 2013]

For professional research mathematicians this sounds too primitive. In fact many don't understand at all was has been said. User Misha, for instance, asked: "This gives you limxØ0 (x - x)/(x - x) = 1. I guess, you are using a system of axioms where 0/0 = 1."

Constantin, a Greek mathematician on sabbatical in Germany, dared to defend my position asking Misha: "Would you disagree that (x - sinx)/(x - sinx) = 1 for every x including the limit? That same holds for tan, arctan and so on? In my opinion Arnold cannot have expected that someone calculates limits. Either you see it - or not."

Few hours later all his contributions were deleted without any announcement and Constantin had been suspended for a month. Later he has been deleted completely. The impression that the "great research-level logicians of MathOverflow" are not so great in sober mathematical thinking must be avoided by all means. Of course also my answer has been deleted. Certainly it was not "research-level".

§ 328 This paragraph is not related to the present MathOverflow series, but I would like to add it here because it is so typical and, for the objective reader, so instructive.

My principle teaching, which I considered trivial and well-understood by every mathematician is this: No infinite sequence can be defined by writing all its terms. We need always give a finite word as the definition from which every term can be calculated. In case of the infinite sequence of digits 0.000... this is done by the three 000 in connection with the three points.

Somebody, the name of whom does not matter, wrote with respect to this explanation: "Everybody understands your trivial crap." However, to the obvious conclusion concerning the path representing 0 in the Binary Tree: "The path 0.000... like every infinite path cannot be defined by nodes", he replied: "Of course it can." So he has not understood and probably never will.

Another reader replied: "Nonsense. If you'd bothered to define a tree properly, it would be clear that a path is defined exactly by the set of nodes in it's edge." Is it possible that he does not see that any proper definition together with his sentence "a path is defined exactly by the set of nodes in it's edge" is not a definition by infinitely many nodes (or terms of the sequences) but a finite word?

Alas, there are only countably many finite words in all speakable and understandable, in short: in all usable, languages together.

§ 329 Here is a question rather as simple as the boys-and-girls-question (cp. § 302). It had existed nearly one year in MathOverflow before I answered it under the pseudonym Hilbert7Problem (which I considered well-received in MathOverflow). http://mathoverflow.net/questions/103816/bike-lock-puzzle

I was wondering this when using my bike lock, a combination lock with four dials, each of which has ten digits (0-9) on it in numerical order. Suppose a bicyclist decides that, from now on, after putting in his combination on this lock, he will only give the lock one twist to close it. So, he chooses between 1 and 4 adjacent dials, and rotates them any number of spaces (other than a multiple of 10, to avoid having the lock end this procedure in a closed position!) Unbeknown to the bicyclist, a thief is following him. The thief knows that the bicyclist uses this procedure to secure his bike. Over a period of days, the thief notes each combination the lock ends up on. What's the fewest observations that the thief needs to make before she can deduce the combination with certainty? What's the fewest observations that she needs to make before she can reduce it to 10 possibilities? How can a shrewd (but stubborn) bicyclist maximize the number of observations necessary without repeating a combination? [Kaveh, 2 August 2012]

If you are clever, the thief needs 49 different settings of the dials to know the correct setting with certainty. This is more than half of all 90 = 4ÿ9 + 3ÿ9 + 2ÿ9 + 1ÿ9 possible settings you can produce (when moving one dial, two adjacent dials, three adjacent dials, and all four dials, respectively, into their nine possible incorrect positions). Let the dials be (a, b, c, d). Let the correct position be (0, 0, 0, 0) to have a mental picture. If you put a in eight different positions, say 1, 2, ..., 8, then the thief does not yet know with certainty the correct one - although she knows the correct positions 0 of the other three. So if you decided to turn one or more other dials leaving a at 0, she would quickly know the complete correct setting. But if you put (a, b), (a, b, c), and (a, b, c, d) also in the eight different positions avoiding a = 9 , you have 4ÿ8 = 32 positions without revealing the correct information. You can do even better, if you choose to start with b. Then you can extend your number of settings by moving (a, b), (b, c), (a, b, c), (b, c, d), and (a, b, c, d) supplying (1 + 2 + 2 + 1)ÿ8 = 48 settings in total. (Of course the order you choose does not matter.) You would get the same opportunity with c instead of b. d however, like a, would supply only 32 possible positions. The maximum number of different positions, before the thief has discoverd the correct one, is for n > 2 digits and an even number m of dials: (n - 2)(m/2)(m + 2)/2 . For an odd number m of dials you get (n - 2)((m + 1)/2)2 . Addition: Maximality If no single dial is moved, we have only 48 settings: 3 pairs, 2 triples and 1 quadruple in 8 positions each. But if pair (a, b) has been moved twice, pair (c, d) cannot be moved without revealing the secret. Hence, we get only 40 settings. That is less than the constructed 48. So, in order to maximize the number of the secret-maintaining settings, we have to move also at least one single dial. But having moved it twice, we can no longer move any other single dial or the pair not containing the first. This subtracts 36 from the 90 possible settings. Since of the remaining 54 settings 6 are always "the nineth", i.e., revealing the secret, we have at most 48 settings. [Hilbert7Problem, 25 June 2013]

First Hilbert7Problem was praised: "This is very nice (and much better than what I wrote)" [S. Carnahan] and the answer got six upvotes. Later Hilbert7Problem dared to point out, very politely though, that the "research-professional's" answer to the boys-and-girls-question (cp. § 302 and §324) shows that at least 100 very stupid users are existing in MathOverflow. Subsequently he was deleted. But my answer to the bike-lock-puzzle remains; only the author has been removed. A copy-right violation.

§ 330 Is there an Oort cloud of inaccessible natural numbers?

Hans Peter asked in MathOverflow: When God, at the end of all time, will check what of his creation has been worthwile, he will also consider the set of natural numbers that ever have been used by his creatures. And he will find that only a very small subset has been applied. (This idea goes back to Borel.) For every usable number we have a finite set of predecessors and an infinite set of ¡0 successors. So there is no usable natural number behind some borderline, although that borderline cannot be determined yet. Is it, in principle, possible to find circumstantial evidence for the existence of the ¡0 inaccessible numbers - in order to satisfy platonists like Gödel? Or is postulating them by the axiom of infinity the only way to lay hold of them?

A first comment by Asaf Karagila: "If time is infinite, then at the end of all time we might have generated all numbers" showed the belief in the end of the not ending as the foundation of matherology, since, if time is infinite, there is no end. We might not have generated all natural numbers at the end of all time, but simply never.

Asaf's second comment showed this again: "Hans, can you imagine any natural number that has infinitely many predecessors? But there are still infinitely many of them. If time is infinite, it suffices that everyone just counts the number of days since today in order to ensure that we used all the natural numbers when time ends."

Hans disproved that there are still infinitely many: "Asaf, I cannot imagine a natural number having infinitely many predecessors. But that is not surprising because the infinitely many numbers always remain successors. The end of all times, however, could not be after a finite number of instants, because then it was finite and was not the end."

§ 331 Hans Peter asked in MathOverflow: Are there undecidable questions that do not depend on the chosen axioms? It has been conjectured that the digit sequence of a normal irrational contains all possible finite digit sequences. Brouwer's famous question after a sequence of nine nines has been answered in the affirmative, thereby weakening a bit his position of reasoning against tertium non datur. Nevertheless there seem to be some arguments against tertium non datur. One of them is this: Will it ever be possible to find out whether the digit sequence of π contains a sequence of 1010000 nines? Of course this question cannot be decided with the current tools. But my question is this: Can anybody imagine a way that would promise an answer in the long range? Or should this problem be considered as undecidable in eternity?

Hans Peter collected 40 reputation points in his short career in MathOverflow, but the ususal gang of deleters appear not to be interested in too obvious arguments that mathematics is not performed in heaven but in our physical reality. Although it can be denied, no computer and no brain and no mathematician can leave it. This situation is comparable to erotics in past times: Man darf das nicht vor keuschen Ohren nennen, was keusche Herzen nicht entbehren können. [Goethe, Faust I]

§ 332 Undefinable numbers

Annix asked in MathOverflow on 29 Oct. 2010: Is the analysis as taught in universities in fact the analysis of definable numbers? http://mathoverflow.net/questions/44102/is-the-analysis-as-taught-in-universities-in-fact-the- analysis-of-definable-numb Ten years ago when I studied in the university I had no idea about definable numbers {{No, such poisonous stuff is not usually taught to normal mathematicians}}, but I came to this concept myself. My thoughts were as follows: - All numbers are divided into two classes: those which can be unambiguously defined by a limited set of their properties (definable) and such that for any limited set of their properties there is at least one other number which also satisfies all these properties (undefinable). - It is evident that since the number of properties is countable, the set of definable numbers is countable. So the set of undefinable numbers forms a continuum. - It is impossible to give an example of an undefinable number and one researcher cannot communicate an undefinable number to the other. Whatever number of properties he communicates there is always another number which satisfies all these properties so the researchers cannot be confident whether they are speaking about the same number. - However there are probability based algorithms which give an undefinable number in a limit, for example, by throwing dice and writing consecutive numbers after the decimal point. {{The limit is never reached by throwing dice or writing something.}} But the main question that bothered me was that the analysis course we received heavily relied on constructs such as 'let's a to be a number that...", "for each s in interval..." etc. These seemed to heavily exploit the properties of definable numbers and as such one can expect the theorems of analysis to be correct only on the set of definable numbers. {{Of course!}} Even the definitions of arithmetic operations over reals assumed the numbers are definable. Unfortunately one cannot take an undefinable number to bring a counter-example just because there is no example of undefinable number, but still how to know that all those theorems of analysis are true for the whole continuum and not just for a countable subset? [Annix, 10:47]

The answer is simple: Analysis is true for all real numbers because there are no undefinable real numbers. But there are many misunderstandings and quibbles - stuff for several paragraphs.

§ 333 Annix's question "Is the analysis as taught in universities in fact the analysis of definable numbers?" (cp. § 332) appeared dangerous to many matheologians, so it got closed after less than 2 hours. It was mainly J.D. Hamkins who pleaded for reopening: I just wrote a long answer to this question, but it was closed just as I was about to click submit. Can we re-open please? I think that there are a number of very interesting issues here. [Joel David Hamkins, 29 Oct. 2010] But the customary gang of closers did not give up! I disagree with the continuing votes to close. [Joel David Hamkins, 30 Oct. 2010] Meanwhile the question has collected 27 upvotes. Obviously the judgement about the quality of questions is extremely subjective in MathOverflow.

§ 334 Annix's question "Is the analysis as taught in universities in fact the analysis of definable numbers?" (cp. § 332) has been reopened and answered by J.D. Hamkins:

In recent work (soon to be submitted for publication), Jonas Reitz, David Linetsky and I have proved the following theorem: Theorem. Every countable model of ZFC and indeed of GBC has a forcing extension in which every set and class is definable without parameters. In these pointwise definable models, every object is uniquely specified as the unique object satisfying a certain property. {{Of course that is not difficult in a countable model. Alas the only interesting property of set theory is uncountability.}} Although this is true, the models also believe {{what models do you allude to, that can believe?}} that the reals are uncountable and so on, since they satisfy ZFC and this theory proves that. The models are simply not able to assemble the definability function that maps each definition to the object it defines. [J.D. Hamkins, 29 Oct. 2010]

Then use only the definition as the "object" that it defines. This has always been done in mathematics with real numbers. None of them is defined by an infinite string of digits or another "object". Each one is a definition or several equivalent definitions like "divide 1 by 9" or "1/9" or "0.111..." (which are not infinite decimal representations). If there is more than one definition, there is no problem, as long as they are equivalent. If you have the choice between several definitions of π, for instance, that does not hurt. If your model can do that, you have a contradiction in ZFC because there is nothing uncountable but uncountability is the main result of ZFC. If not, it is useless.

§ 335 Annix's question "Is the analysis as taught in universities in fact the analysis of definable numbers?" (cp. § 332) has raised some comments by Andrej Bauer: He {{J.D. Hamkins}} did not say that it is consistent to "postulate in ZFC that undefinable numbers do not exist". What he was saying was that ZFC cannot even express the notion "is definable in ZFC". {{Therefore it is very surprising that ZFC is allegedly able to define sets and numbers, i.e., to do things that nobody in ZFC can prove to be done correctly.}} Joel made a very fine answer, please study it carefully. Joel states that there are models of ZFC such that every element of the model is definable {{although nobody can know precisely what that means}}. This does not mean that inside the model the statement "every element is definable" is valid. The statement is valid externally, as a meta-statement about the model. Internally, inside the model, we cannot even express the statement. {{And externally we cannot find out whether external statements are meaningful, because "external" is also only some model - yet a bigger one.}}

Annix answered: I do not say undefinable numbers do not exist. Their existence follows from axiom of choice {{perhaps it follows, but as a contradiction, because you cannot choose one of many undefined numbers}} and in theory we can uniquely define each undefinable number by specifying infinite number of its properties. The problem is that the theorems of analysis as taught in universities sufficiently rely on the properties of definable numbers. {{That is not a problem of mathematics since other numbers have no properties. Also it is not a "problem" of ZFC but a simple contradiction in ZFC. Of course Zermelo would not have been stupid enough to defended in his 1908 paper the axiom of choice in length as a natural choice if he had been confronted with undefinable numbers. That nonsense has only become en vogue in the circles of modern "logicians". Of course nobody can say what real number is undefinable. Why don't the undefinable-number-cranks believe in undefinable natural numbers? Of course nobody can say what natural number is undefinable. But then countability-spook would no longer haunt those poor peolple's mind.}}

§ 336 Annix's question "Is the analysis as taught in universities in fact the analysis of definable numbers?" (cp. § 332) has raised another comment by Andrej Bauer: This is off-topic, but: it makes no sense to claim that "constructivist continuum is countable in ZFC sense". What might be the case is that there is a model of constructive mathematics in ZFC such that the continuum is interpreted by a countable set. {{Why then not use this as the model of university mathematics and drop all blather about uncountability?}} Indeed, we can find such a model, but we can also find a model in which this is not the case {{this is no contradiction, of course}}. Moreover, any model of ZFC is a model of constructive set theory {{and constructive models contraditct uncountability because everything constructed, say as a constructed anti-diagonal of a constructed Cantor-list, is well-defined and therefore definable and therefore belonging to a countable set}}. You see, constructive mathematics is more general than classical mathematics, and so in particular anything that is constructively valid is also classically valid {{for instance the theorem that uncountability does never occur constructively}}.

§ 337 Annix's question "Is the analysis as taught in universities in fact the analysis of definable numbers?" (cp. § 332) has raised a comment by arsmath: "Definable numbers" are numbers that are definable in terms of first-order logic over set theory. There are perfectly intelligible numbers that cannot be defined in your sense. For example, suppose you have a sequence of definable numbers an that is bounded by a constant. Then b = sup an is a number that is unique and has an unambiguous meaning, but b is not necessarily definable. {{It is defined by the sequence (an). Why should another definition be searched? Irrational numbers are never defined in another way since there is no irrational number that can be defined by giving the value of every digit. In Excel you are often asked whether you wish to copy only values or also formulas. If you answer "only values" you wil never get an irrational number.}} Each an is given by a formula {{of course, real numbers cannot be given in another way}}, but if the formulas are sufficiently different then there is no way to write down a single formula for b {{then it is already impossible to write down all formuas for the an. If the sequence should be defined completely, then there must be a first n from which all following an are defined by the same formula. Otherwise you'd have to write down infinitely many formulas, which is as impossible. In that case there is not an undefined b but no b at all.}}

Annix answered in this spirit: Yes, b is not definable. But it is also not unique for any bounded number of properties we define (i.e. bounded number of sufficiently unique ϕn(x)). Thus for each limited number (say, N) of ϕn(x) we get infinitely many numbers which satisfy for first n < N. arsmath replied: I'm not clear what you're after. You said that a researcher cannot give an example of an undefinable number, and that one researcher cannot communicate an undefinable number to another. I pointed you towards a counterexample to both claims. {{And I pointed you to the error you commited.}} You can give a completely explicit family of formulas, so explicit that they can be generated by a computer program, that gives you a number that's not definable. We can't say much about that number, but it still have a description that identifies it uniquely {{and thus defines it}}. Most theorems of analysis that are false if you only consider definable numbers. For example, the set of definable numbers does not have the least upper bound property. The intermediate value theorem is false, etc. {{and undefinable numbers do not in the least improve the situation, because only definable numbers are looked for in these theorems.}}

Finally, J.D. Hamkins told asrmath the same as I said here: Arsmath, you haven't actually described a non-definable number, and it is impossible to do so for the reasons expressed in my answer. {{Yes, that is correct. And therefore the undefinable numbers are without any worth in mathematics - and elsewhere.}}

Annix took the same position: Either the family of formulas is finite and can be communicated to the other researcher, then the number b is definable. Or it is infinite, then it cannot be communicated to the other researcher. You actually did not give an example of b since you say nothing about the defining formulas. Thus b is not defined so far. {{Correct.}} arsmath had not yet understood, but this paragraph has become very lengthy already. Therefore this very instructive discussion will be continued in the next paragraph.

§ 338 The problem of understanding the meaning of definability is basic to modern set theory. Therefore the discussion triggered by a question of Anixx in MatheOverflow http://mathoverflow.net/questions/44102/is-the-analysis-as-taught-in-universities-in-fact-the- analysis-of-definable-numb is continued here: arsmath: My point is that you can write down an explicit description of a number that unambigiously defines it to a human being, but that number is not definable. {{On the contrary. This number is defined.}} I can (in theory) provide a finite computer program that provides the formulas. (The Wikipedia page sketches a similar construction under "Notion does not exhaust...".) {{Wikipedia is written by humans. Humans often fail.}} This is a philosophical question {{not at all}}, but I would say a finite computer program is sufficient. How do I know that 100001000010000 exists? {{Because you have written it here.}} Because I can write a computer program to compute it. {{No computer could do more than you have done here. In particular every computer would fail to count in single steps up to 10100.}}

Anixx: In that case your number b is not only definable but even computable. arsmath: It's not computable because the individual ϕi are not necessarily computable.

Anixx: In that case it is not computable but definable. arsmath: Why? Definability requires a single formula.

Anixx: If there is a program that can generate such formulas, then there is a single formula for all. {{Of course.}}

A. Blass: Your last comment is based on the erroneous assumption that one can define how to pass from a definition to the thing it defines. {{No, the definition of an immaterial thing is the thing it defines.}} Arsmath described a situation where a sequence of formulas might be definable (and even computable) but the sequence of real numbers they define is not definable. Joel explained why your assumption is wrong. {{Joel explained why his assumption is right.}} I second Andrej's earlier suggestion that you study Joel's answer carefully, and I add my own suggestion that you assume that Joel meant exactly what he said, not what you think he should have meant or must have meant. {{What did Joel say? "Arsmath, you haven't actually described a non-definable number, and it is impossible to do so for the reasons expressed in my answer." I would recommend that you study this answer very carefully, A. Blass!}}

Anixx answered on the erroneous assumption mentioned by A. Blass: "please tell me where I did such assumption (and what do you mean under "pass")? Arsmath described a situation where a sequence of formulas might be definable (and even computable) but the sequence of real numbers they define is not definable. - It may be not computible but how it can be undefinable? Can you give an example? {{No, of course he cannot and so he did not since by definition undefinable numbers are not definable.}} And, by definition, undefinable number is such number that for any limited set of its properties there is at least one other number with the same properties. {{So it is!}}

Andres Caicedo: It feels like we are going in circles now. {{No, it feels that some people in fact have recognized that undefinable numbers are not definable.}}

Finally, some years later, I entered the discussion that I had not been aware of before: It is completely irrelevant how to pass from a definition to the thing it defines. Important for the present countability question is only the correspondence of definition and defined object. In mathematics real numbers and their definitions are in this correspondence: There are many definitions for some individual real numbers like e, but there are no definitions of individual numbers that fail to define individual numbers.

§ 339 Annix had asked in MathOverflow on 29 Oct. 2010: Is the analysis as taught in universities in fact the analysis of definable numbers? I gave the following answer (that of course has been deleted soon): A definable real number r is a number that can be defined, i.e., r can be identified and communicated by a finite sequence of bits in real life, just where mathematics takes place. This makes the set of simultaneously (in a given language) definable numbers countable. Therefore all real numbers that can appear in the language of mathematical analysis belong to a countable set. Independent of real-life conditions it is impossible to distinguish, in the universe of ZFC or elsewhere, real numbers by infinite sequences of bits. This claim is proven by the possibility to construct all infinite sequences of digits by means of a countable set of infinite sequences of digits as follows: Enumerate all nodes ai of an infinite binary tree and map them on infinite paths pi such that ai is in pi. There is no further restriction. The mapping need not be injective. Then construct from this countable set of paths another binary tree. Mathematical analysis is not able to discern which paths were used for construction. This shows that outside of a platonist ZFC-universe there are not uncountably many real numbers. Real numbers created by Cantor-lists are not defined unless the Cantor-list is well- defined, i.e., every entry of the list is known. That requires a Cantor-list constructed by a finite definition. But there are only countably many finite definitions of Cantor-lists. The existing real numbers of analysis cannot be listed. But that does not make their set larger than any countable set.

§ 340 carl-labande asked in MathOverflow: What is the fade-away-rate of mathematical induction in practical applications? The unexpected hanging paradox, hangman paradox, unexpected exam paradox, surprise test paradox ... All these paradoxes and many others, like the blue-eyed islanders paradox, are mainly based on the unlimited validity of induction. But perhaps this assumption is incorrect with respect to application of mathematical induction to practical reality. If the teacher announces "tomorrow we will write an unexpected exam", then this is clearly a self-contractory announcement, even in reality. If the teacher announces that the unexpected exam will be written next week or in any specified interval of days, then many mathematicians tend to conclude that this is also self- contradictory in reality because induction shows that the last day of the interval cannot apply, therefore also the day before the last one cannot apply, and so on. But for a really long interval induction fails as can be proved. Consider that the teacher announces one or even 100 surprise tests during the next 3000 days, then induction won't help at all to determine the dates. In order to prove that, guess 100 dates and compare with a set of 100 random numbers of that interval. That suggests: in these cases the reasoning based upon induction does not remain valid for large intervals in reality. (Compare the blue-eyed islanders paradox with 1010 islanders.) The validity of is certainly absolute for n = 1, but near to zero for n = 1010 and has limit 0 for an infinite interval. The question is of course, whether this problem belongs to mathematics or to reality only. But I would plead the case that also problems with importance for reality should be scrutinized by mathematicians. Therefore my question: What is the fade-away-rate? Can a "function of validity" f(n) be defined concerning the state of knowledge at time zero?

The question you ask is not a mathematical one. While the question might be research-worthy, the research in question is not mathematical. [Boris Bukh]

You may be right. But my question is just whether this problem has been considered in mathematical literature. I guess this is possible. Why should mathematicians refuse to help in practical questions? Perhaps mathematical methods will be useful to find such a function? [carl- labande]

Mathematicians might well love to help in a practical question. They just might not have a clue how to do so. [Lee Mosher]

Yes, I also made this observation. If a question appears too difficult or if they cannot get to the interesting nucleus, soon some dwarf-mathematicians gather together and close the disturbing proof of their inability {{cp. Annix's question, § 332}}. The present question was closed by Boris Bukh, Gerald Edgar, Steven Landsburg, Felipe Voloch, and Qiaochu Yuan.

As with Anixx's question it was again J.D. Hamkins who discovered some interesting aspects here: "Suppose you only knew that the exam would be held and would be a surprise with certain probabilities, perhaps very high. Then one might hope to propagate the inductive reasoning through this probability, perhaps ultimately giving a probability distribution for when the exam would occur? Can someone give an answer along these lines?" But in this case he lost against the dwarfs.

§ 341 Recently Peter Shor's community-wiki-question in MathOverflow: "Can a mathematical definition be wrong?" http://mathoverflow.net/questions/31358/can-a-mathematical-definition-be-wrong has been answered by Hans Peter: Definition: "This is a wrong definition." If definitions cannot be wrong, then this is truely a wrong definition. If definitions can be wrong, nothing remains to be shown.

§ 342 gowers asked on 3 April 2012: What was Gödel’s real achievement? http://mathoverflow.net/questions/20219/what-was-godels-real-achievement When I first heard of the existence of Gödel's theorem, I was amazed not just at the theorem but at the fact that the question could be made precise enough to answer: how on earth, even in principle, could one show that it was impossible to prove something in a given system? That doesn't bother me now, and that is not my question. It seems to me that Gödel's theorem is a combination of at least three amazing achievements, namely these. 1. Formalizing the notions of proof, model, etc. so that the question could be considered rigorously. 2. Daring to think that there might be true but unprovable statements in Peano arithmetic. 3. Thinking of the idea of Gödel numbering and getting the proof to work. ...

My answer was that none of the three points was Gödel's real achievement: I think Gödel's real achievement has been overlooked grossly, perhaps because it only appeared in a footnote. He recognized and wrote in his seminal paper [1] that the true reason for the incompleteness is caused by the transfinite hierarchy: "Der wahre Grund für die Unvollständigkeit, welche allen formalen Systemen der Mathematik anhaftet, liegt [...] darin, daß die Bildung immer höherer Typen sich ins Transfinite fortsetzen läßt [...] während in jedem formalen System höchstens abzählbar viele vorhanden sind. Man kann nämlich zeigen, daß die hier aufgestellten unentscheidbaren Sätze durch Adjunktion passender höherer Typen (z. B. des Typus ω zum System P) immer entscheidbar werden. Analoges gilt auch für das Axiomensystem der Mengenlehre." So, if Brouwer is right with his statement excluding the transfinite hierarchy: "De tweede getalklasse van Cantor bestaat niet" [2], then Hilbert's program [3] gets support from an unexpected side and can be restarted. [1] Kurt Gödel: "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I", Monatshefte für Mathematik und Physik 38 (1931) S.173–198, quoted from p. 191. [2] L.E.J. Brouwer: "Over de grondslagen der wiskunde" (Februari 1907, Dutch) Thesis XIII. http://www.archive.org/details/overdegrondslag00brougoog [3] E. Artin et al. (ed.): "D. Hilbert: Die Grundlagen der Mathematik" (1927). Abh. Math. Seminar Univ. Hamburg, Bd. 6, Teubner, Leipzig (1928) 65-85. English translation in J. van Heijenoort: "From Frege to Gödel", Harvard Univ. Press, Cambridge, Mass. (1967) 464-479. http://www.marxists.org/reference/subject/philosophy/works/ge/hilbert.htm [Wolfgang Mueckenheim]

I doubt that gowers has seen my answer because the usual gang deleted it soon. Gödel, Brouwer, and Hilbert, put together in an unorthodox way, seem to be considered a dangerous triumvirate.

§ 343 On 11 June 2013 Ari asked in MathOverflow: Can machines generate truly random sequences? http://mathoverflow.net/questions/133388/random-infinite-sequence-can-machines-generate- truly-random-sequences Hans Peter answered: Yes, they can. Couple a computer with a Geiger counter that clicks between 100 and 1000 times a minute. Count the beeps. If the number is even, let the computer print 0 if it is odd 1. The sequence is absolutely random. {{Alas, this is not matheology but MatheRealism. Therefore his answer has been deleted soon.}}

§ 344 Where mathematics has gone wrong

Alekk asked in MathOverflow: I would be interested in knowing examples of results conjectured by physicists and later proved wrong by mathematicians. Furthermore it would be interesting to understand why physical heuristics can go wrong, and how wrong they can go (for example, were the physicists simply missing an important technical assumption or was the conjecture unsalvagable). http://mathoverflow.net/questions/30149/examples-where-physical-heuristics-led-to-incorrect- answers

I answered: Why do you ask for wrong physicists only? Being wrong happens to mathematicians as well. In 1833, the year of his dead, Adrien Marie Legendre presented an overwiev of proofs of the parallel axiom to the French Académie des Sciences. It included six rigorous proofs, three of which using infinite angular areas. (Here "rigorous" is to be understood in the meaning of his times as present mathematicians use "rigorous" in the meaning of our times. But obviously there can never be absolute rigour, neither then nor today.) Or take the first proof of the Cantor-Bernstein theorem bei E. Schroeder in 1896. The proof was wrong, as Schroeder admitted in a letter to A. Korselt (who had improved the proof). Korselt gives a copy of Schroeder's reply in his paper [A. Korselt: "Über einen Beweis des Äquivalenzsatzes", Math. Ann. 70 (1911) 294.] Nevertheless, Korselt's corrected version was not accepted in 1902 by the Annalen. Only 9 years later, he could publish his paper. But that was not widely noticed, so the incorrect proof survived for a long time. Cantor wrote to Hilbert on June 28, 1899 that E. Schroeder in 1896 (and Cantor's student F. Bernstein about Easter 1897) had proved the theorem. So Cantor never noticed Schroeder's error. A. Fraenkel mentioned in 1923 (!) that Schroeder's proof was wrong. [A. Fraenkel: "Einleitung in die Mengenlehre", Springer (1923) p. 58] E. Zermelo considered Schroeder's proof correct even in 1932. Zermelo remarks in Cantor's collected works as an editing note: "... wurde erst im Jahre 1896 von E. Schroeder und 1897 von F. Bernstein bewiesen und seitdem gilt dieser 'Aequivalenzsatz' als einer der wichtigsten Saetze der gesamten Mengenlehre." [E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer (1932) p. 209] This shows that wrong things can survive in mathematics for about 35 years. Or even longer? And should that be different now?

§ 345 Where Cantor as a physicist has gone wrong

After this answer had been deleted, I gave a second one: Georg Cantor, the founder of set theory, also gave lessons in philosophy and in theoretical physics. He devised his set theory in particular to get a better explanation of physical phenomena. Therefore he can be considered as a physicist, at least in part. He stated that "in the universe and on earth and, according to my firm conviction, in every non vanishing volume of space there are an actually infinite number of created creatures." [Letter to Cardinal Franzelin of Jan. 22, 1886] Cantor believed that the material point-like atoms were a countable set and the ether atoms were an uncountable set (though he did not believe in the existence of atoms for chemical purposes). Both sets should be dense (in sich dicht) and geometrically homogeneous. [Letter to Mittag-Leffler, Nov. 16,1884] These ideas have later been proven incorrect. (By other physicists. Sorry, again not fully met the topic.)

Of course also this answer of mine has been deleted, as was to be expected in this forum of pompous Cantor-admirers and -victims.

§ 346 How can undefinable objects be elements of mathematics?

MaBru asked recently, on Friday, 13 Sept. 2013 (obviously not a very lucky date) in MathOverflow: How can undefinable objects be elements of mathematics?

Consider the complete infinite Binary Tree. It has countably many nodes to be enumerated as follows: 0 / \ 1 2 / \ / \ 3 4 5 6 / \ / \ / \ / \ 7 8 9 10 1112 1314 ... Remove the nodes one by one and with each node remove a path containing that node. If the first nodes of a path are already missing, remove only the remaining tail of the intended path. First remove node 0 and, for instance, the path (0, 1, 3, 7, ...) Then remove node 2 and, for instance, the remaining nodes (2, 6, 14, ...) of the path (0, 2, 6, 14, ....) Then remove node 4 and, for instance, the remaining nodes (4, 10, ...) of the path (0, 1, 4, 10, ...). Continue until all countably many nodes and contably many paths have been removed. If there are uncountably many paths in the Binary Tree, then they cannot be defined by nodes, since there are no nodes remaining. With respect to the structure of the Binary Tree, obviously countably many nodes cannot define more than countably many paths. Those paths can only be defined by finite definitions. In fact each path is defined by a finite definition like "(0, 1, 3, 7, ..."). Alas there are only countably many finite definitions in all usable languages. How can undefinable objects be elements of mathematical discourse? [ MaBru]

The problem with the view that "every path has to be defined by a finite definition" is that as soon as you specify what counts as a definition, you open a possibility to specify (by diagonalization) a path which is not (by your standards) definable. Why should that path not be an element of mathematical discourse? [Johan Wästlund]

Of course that path is an element of mathematical discourse, since it belongs to a countable set of paths. Every defined diagonal of any defined list belongs to the countable set of defined elements of mathematical discourse. {{In fact a "problem with the view" results only if one insists that infinity can be finished, that there is a list of all natural numbers. But who would tolerate such an Overflow of Madness?}} A further question. Perhaps the closers can answer, but probably they cannot and therefore will prefer to delete this question: What views exist in mathematics besides "the view that 'every path has to be defined by a finite definition'"? [MaBru]

MaBru appears clairvoyant. Instead of an answer the question got deleted.

§ 347 Countability and super tasks

Recently I have been convinced that enumerating the rational numbers is a super task: Counting the positive rational numbers can be interpreted as a super task: In the nth step fill into an urn all rationals between n-1 and n and, if not yet residing in the urn, also the rational number qn to be enumerated by the natural number n. Take off the rational number qn. Go to step n+1. But, in spite of ardent prayers, I cannot believe that such super tasks can be finished with the result "urn empty". Further I don't know a suitable axiom. Therefore my question: Can this result be circumvented? Can someone supply a convincing argument that the enumeration of the rational numbers is not a super task? [superintendent]

Comments after some hours: Nobody wiling to help me? [superintendent] Nobody able to help me? [superintendent] You should find another web site that is more appropriate for your question. It might help if you understood the difference between defining a map on infinite sets and doing a supertask. [S. Carnahan] What is the difference between defining a map on infinite sets and doing a supertask? That's what I wish to know! But you are not willing to say in two words what the difference is? Can you point me to someone who would know it??? [superintendent]

Of course this dangerous question could not persist in MathOverflow; it was deleted immediately; no traces remain. Perhaps someone is able to figure out the difference?

§ 348 Three bawdy points

When discussing the super task of § 347 in MathOverflow it is generally regarded as unmathematical and unprofessional and punished by deletion to mention one of the following three points. It appears like mentioning genitals in the presence of ladies in Queen Victoria's times!

1) Do you properly distinguish every and all? Every number has finitely many predecessors and infinitely many successors. In mathematics you can easily conclude that every natural number belongs to a set that contains less than half of all natural numbers because "infinitely many" is larger than "finitely many".

2) When the urn is empty in the end of the unending, it contains no rational number. Do you agree that a stringent property of the super task requires that in every step only one number leaves the urn? If there is a state with none, then there must have been a preceding state with one. Or do you simply neglect this property?

3) Can you tell me why Cantor's bijection is far from being a super task? The above delusions are simply nonsense because ...... (please fill in your argument).

§ 349 Pete L. Clark, addressing me, wrote in MetaMathOverflow in July 2010 http://tea.mathoverflow.net/discussion/484/physicists-can-be-wrong/ what I will comment here:

P.L. Clark: I am not sure that you understand the technical items of set theory. ... You use terms like completed versus potential infinity, which are not part of the modern vernacular.

WM: You are not well informed. But partially you are right. Many set theorists do not like to talk about finished infinity. These words have been omited from the official vocabulary with good reason: An intelligent newbie would get somewhat confused. But what you formalize in modern set theory is just completed infinity - even if you don't know that.

P.L. Clark: ... try out your argument on the following simpler case: consider the infinite graph on the integers where for all n, n is adjacent precisely to n-1 and to n+1. There are countably many nodes on this graph but there are uncountably many random walks: they again correspond to infinite sequences from a two-element set. This is disturbing to you because...?

WM: This is disturbing because there is no chance to have an infinite random walk executed, described, or used in mathematics other than by a finite description. Alas, there are only countably many finite descriptions in all usable languages, i.e., in all languages that can be learned, written and read. (An uncountable alphabet would be useless, because an alphabet is something to look up. But an uncountable alphabet cannot fit into a list - not even into the whole universe. Therefore it is a contradictio in adjecto. I would never get in touch with people who seriously consider uncountable alphabets because they must be so much confused that they are probably even dangerous.)

P.L. Clark: Let's not be disingenuous: you are notorious on the internet for your writings about set theory and especially Cantor's uncountability arguments. But Cantor's work on set theory has been explored and vetted with extreme care by mathematicians for more than a hundred years.

WM: Same has been done by (even more) astrologians with astrology.

P.L. Clark: Nowadays our attitude to allegations of flaws in Cantor's work is similar to that of many biologists when presented with attacks to evolution from "creation scientists":

WM: Crazy! It is just the other way round. Cantor's first proofs of finished infinity were based on God and the Holy Bible. He coined his expression cardinal number when he tried to convince cardinal Franzelin of his absured ideas in a letter of 22 Jan. 1886. And he was a hard core creationist with respect to nature as well as to mathematics. He even tried in several cases, by intrigues, to keep Darwinists from university chairs. Always without success, fortunately.

P.L. Clark: ... it is not a debate we are eager to have, and we feel that we are at least entitled to restrict ourselves to discussants who show an understanding and technical mastery of the relevant material (which is, for mathematics, not that technical: for instance, many bright high school students know it well).

WM: Sorry, due to my professional responsibilities I often have to meet bright high school students but I never met a high school student who was informed that modern matheology requires to accept undefinable objects as real numbers. But when I told some bright students about that "fact", they spontaneously asked what a Cantor-list of undefinable numbers and its diagonal number would look like.

P.L. Clark: There's certainly room for philosophical doubts about uncountable (or even countably infinite) sets, but this is not the appropriate forum for that.

WM: In particular since my arguments are not philosophical ones but simply mathematics. (A summary will be given in § 350.)

P.L. Clark: From our point of view, your criticisms are simply not valid.

WM: I know. That will never change. But perhaps I will save some young students from getting matheologians. I'll do my best.

§ 350 The Four Most Simple Contradictions of Transfinite Set Theory

I started to learn about large cardinals a while ago, and I read that the existence, and even the consistency of the existence of an inaccessible cardinal, i.e. a limit cardinal which is additionally regular, is unprovable in ZFC. Nevertheless large cardinals were studied extensively in the last century and (apart from attempts that went too far as the Reinhardt-Cardinals) nobody ever found a contradiction to ZFC. [erinna, 29 Oct. 2010] http://mathoverflow.net/questions/44095/arguments-against-large-cardinals

The reason seems to be that all contradictions are eagerly and quickly deleted. Today I found this post by WM in sci.math: The actually infinite is self-contradictory as has been shown by many proofs. I will sketch here only four of them: 1) If actual infinity exists, then you can in ¡0 steps well-order the rational numbers by magnitude. 2) A Cantor-list that is complete with respect to the rationals contains every finite initial sequence of the anti-diagonal infintely often. If a list contains every finite initial sequence of π, then it contains π because there is not more than every finite initial sequence. Why? See here: 3) The Binary Tree that contains only all rational numbers of the unit interval contains also all irrational numbers of the unit interval although none of them has been added explicitly. Only countably many rational numbers have been added. But when ready, abracadabra, the Binary Tree contains uncountably many irrational numbers. 4) The sequence 1 2, 1 3, 2, 1 ... contains Ù in every column. So the whole matrix contains Ù. But none of the lines contains Ù. On the other hand, the matrix is constructed such that a set of finite lines never contains more than one of them. And the matrix contains only finite lines. (How many of them does not matter and does not change the facts.) If only one of these ideas is correct {{in fact all are}} then there is no cardinal number larger than infinity. Isn't this an argument against large cardinals? [Hans Peter, MathOverflow, 10 June 2013]

§ 351 An air of unreality

When modern set theory is applied to conventional mathematical problems, it has a disconcerting tendency to produce independence results rather than theorems in the usual sense. The resulting preoccupation with "consistency" rather than "truth" may be felt to give the subject an air of unreality. [Saharon Shelah: "Cardinal arithmetic for skeptics", arXiv:math/9201251 (1992)]

An air of unreality? Very solid air. Rather ore.

§ 352 Colours

Theorem. The set Ù of all natural numbers does not exist. Proof. Let us assume that the actual infinity ¡0 or the set Ù of all natural numbers can exist. Also assume that there exists a set of ¡0 different colours as {red, yellow, green, blue, ...} (the wavelength-range may be as large and differences in wavelength may be as small as desired). We represent the set of all natural numbers by use of the initial segments of colours encoded by r, y, g, b, ... r ry ryg rygb ... The first horizontal line shows number one, the second number two, and so on. It is obvious that the whole set of ¡0 different colours (or the greatest initial segment) will not be used in any horizontal line (because of the finiteness of every horizontal line). In other words the number qayn {{the 22th letter of the Persian alphabet, the first letter of the word "impossible"}} of different colours used is less than ¡0. On the other hand, the number qayn of all different colors used is larger than every finite number. So the following relation holds: Every finite number n < qayn < ¡0 Since every number less than ¡0 is a finite number, we have qayn is a finite number and qayn is larger than every finite number: qayn < qayn This deduction is a contradiction. And now the proof is completed. Therefore the existence of the set Ù of all natural numbers is impossible. [S.S. Mirahmadi (Sept. 2013), Qom seminary, Qom, Iran] [email protected]

§ 353 Hessenberg's proof is this: If a mapping from Ù to all its subsets would exist, then there must one natural number n be mapped on that subset Sn that contains only natural numbers that are mapped on subsets which do not contain the natural numbers which are mapped on them. But does this subset Sn exist? If the set of 2¡0 subsets of the natural numbers would exist, i.e., if Hessenbergs proof would be valid, then one should expect that also all permutations of the natural numbers would exist and (by the bijection of Ù and –) one should further expect that also all permutations of the rational numbers, each rational number indexed by a natural number, should exist. Each permutation is a well-ordering. One of them would be the well-ordering of – that is simultaneously the well- ordering by size. This is a contradiction. Like Hessenberg's assumption.

§ 354 The Continuum C (or the field R) appears as a numerical approximation to a complex reality of observations. - It is not a set in the original sense of traditional set theory. In particular, the power set axiom cannot be applied to it. [...] the entaglement phenomenon involving light (or electrons), discovered by Albert Einstein, 1935, John Stewart Bell, 1964, and Alain Aspect, 1982, demonstrates that the Continuum of light is interiorly somehow “tightly interlaced”, so that the distance, even enormous, between its “entangled” points becomes unimportant; (3) this signifies that the Continuum is not a set, a “bag of points”, but that the points on it appear as the consequence of our activities. [Edouard Belaga: "From Traditional Set Theory – that of Cantor, Hilbert, Gödel, Cohen – to Its Necessary Quantum Extension", Institut des Hautes Études Scientifiques (2011)] http://preprints.ihes.fr/2011/M/M-11-18.pdf

§ 355 The preface {{of my book "Mathematik für die ersten Semester" https://portal.dnb.de/opac.htm;jsessionid=CBA7F11D6BADF59FACD934DA2F83CE82.prod- worker4?method=showFullRecord¤tResultId=Wolfgang+M%C3%BCckenheim%26any&c urrentPosition=1 }} suggests that the book lays a solid foundation for, amongst other things, computer science. {{That is correct.}} Presumably in WMaths, the thing that is not exactly a set of real numbers is countable, as is the thing that is not quite a set of computable numbers. {{Meanwhile it should be clear that "countable" with respect to infinite is a self-contradictory notion. Why not simply say infinite?}} Apart from simply being wrong {{How that? Today everybody can be enlightened enough to know that a set with more than countably many elements is simply the delusion of a small minority of lunatics who adhere to Cantor's transfinity which simply has no place in any rational mind. Even if God had created an uncountable set, no human could apply more than countably many of its elements. No mathematical result, no question or equation could apply or ask for an undefinable real number. But mathematics is what mathematicians do - not what matheologians counterfactually believe.}}, it raises the question of whether or not they are the same. Only WM knows. {{The book says that Ù is a proper subset of —.}} [Ben Bacarrise, "§ 350 The Four Most Simple Contradictions of Transfinite Set Theory", sci.logic (26 Sept. 2013)]

§ 356 It seems likely that only in the realm of pure mathematics can the idea of infinity be entertained. In the context of actual, manifest, realisable quantities things seem much more like the situation in a computer where all phenomena have definite resolution and size. One can never create an infinitely large file because that would require an infinite amount of time and infinite computational resources such as memory. In my own work, which uses computational concepts to model reality I take the position that the phenomena can be arbitrarily large and detailed but they always have a definite finite value. So this allows for potential infinity but totally disallows actual infinity. Given that any set must be actually represented using data (e.g. binary data), then no set can be infinitely large and if one removes any members of the set then the cardinality (size) of the set is reduced. So any representable set cannot be an infinite set and any infinite set cannot be actually represented. Furthermore, in the context of computational metaphysics, representation is equivalent to existence. If something is represented and it takes part in the overall simulation of the universe then it exists in that universe but if it cannot be represented then it cannot exist. So if actual infinities exist then there cannot be any discrete computational foundation to reality but so far no actual infinites have ever been discovered. Even with the domain of pure mathematics, infinities can only exist because they are symbolically represented and never actually represented. No one has ever written out an infinite number of integers thereby actually representing the set of integers. It is only ever referred to but never fully represented. If one required sets to be fully represented then mathematics could not operate on actual infinite sets; it could only operate on potentially infinite sets which always have finite representations (e.g. {1, 2, 3}) but which are unlimited in their length. Such sets are arbitrarily large but always have a definite finite size. [John Ringland: Does Infinity Exist?] http://www.anandavala.info/TASTMOTNOR/Infinity.html

§ 357 Nonstandard mathematics shows new ways of making mathematical discourse more intuitive without losing logical rigor and giving more flexible ways of constructing mathematical objects. We may say that by discriminating between "actual finiteness" and "ideal finiteness", we obtain a better system of handling infinity than the "actual infinity" offers.

The followings are some of the features of our approach radically different from the usual mathematics. Sets are finite. The usual "infinite sets" such as Ù and – are considered as proper classes so that the totality is not considered as a definite object. Sorites Axiom. A number x is called accessible if there is a certain concrete method of obtaining it. We postulate the existence of inaccessible numbers as the most basic axiom of our framework. The accessible numbers form an nonending number series which is closed under the operation x Ø x+1 but differes from the total number series. Accordingly, fundamental notions such as transitivity, equivalence relation, provability, compatibility, etc. become relative to the number series chosen. The overspill axiom. If an objective condition holds for all accessible numbers, then it holds also for an inaccessible number. Here a condition is called objective if it can be specified without the notion of accessibility. Vague conditions. The vaguness of the accessibility prohibits us to regard the collection of accessible numbers as a set. It is a proper class contained in a finite set, called semiset in Alternative Set Theory of Vopenka. Continua are not infinite sets. The real line is considered as the "quotient" of the proper class – by the indistinguishability relation defined by r º r' if and only if k|r - r'| < 1 for every accessible number k.

[Toru Tsujishita: "Alternative Mathematics without Actual Infinity", arXiv (2012)] http://arxiv.org/pdf/1204.2193v2.pdf

§ 358 Das Unendliche und die Theologie / Infinity and Theology (1)

Für die breite Masse gilt Adam Ries(e) als größter (und häufig auch als einziger) deutscher Mathematiker. In bildungsnäheren Schichten nimmt Carl-Friedrich Gauß diese Position ein. Die Mathematiker selbst aber verehren ihren größten Kollegen in Georg Cantor. Die höchste Auszeichnung der DMV trägt sein Konterfei und seinen Namen, denn er hat die Mathematik unendlich erweitert und bereichert --- so glauben jedenfalls die meisten. Über Georg Cantor, Schöpfer der Mengenlehre und Gründer und erster Vorsitzender der Deutschen Mathematiker- Vereinigung, wurden mehr biografische Notizen gesammelt und veröffentlicht als über jeden anderen Mathematiker des 19. Jahrhunderts. Aus diesen Mosaiksteinen lässt sich ein plastisches Bild seiner Weltsicht zusammensetzen. Einige Aspekte, vor allem theologischer Natur, die zu seinem Verständnis des Unendlichen führten und dies scheinbar untermauerten, sollen in den folgenden Beiträgen nachgezeichnet werden.

The educationally disadvantaged populace admires Adam Ries(e) as the greatest (and often as the only) German mathematician. For the educated class Carl-Friedrich Gauss assumes this position. The mathematicians themselves however admire Georg Cantor as their greatest colleague. The highest award of the German Mathematical Union (DMV) carries Cantor's likeness and name, because he extended and enriched mathematics infinitely - at least many believe that. More biographical material has been collected about Georg Cantor, inventor of set theory and first president of the DMV, than about any other mathematician of the 19th century. From these tesserae we can obtain a vivid picture of his world view. In the following paragraphs some of the theological aspects which lead to and seemingly supported his understanding of the infinite will be reproduced. The translations into English are mine. The German originals can be seen in de.sci.mathematik.

§ 359 Das Unendliche und die Theologie / Infinity and Theology (2)

Gestatten Sie mir aber dazu zu bemerken, daß mir die Realität und absolute Gesetzmäßigkeit der ganzen Zahlen eine viel stärkere zu sein scheint als die der Sinnenwelt. Und daß es sich so verhält, hat einen einzigen, sehr einfachen Grund, nämlich diesen, daß die ganzen Zahlen sowohl getrennt wie auch in ihrer actual unendlichen Totalität als ewige Ideen in intellectu Divino im höchsten Grade der Realität existiren. [Cantor an Hermite, 30. Nov. 1895]

Allow me to remark that the reality and the absolute principles of the integers appear to be much stronger than those of the world of sensations. And this fact has precisely one very simple reason, namely that the integers separately as well as in their actually infinite totality exist as eternal ideas in intellectu Divino in the highest degree of reality.

§ 360 Das Unendliche und die Theologie / Infinity and Theology (3)

Hochverehrter Pater Ign. Jeiler. Es freut mich sehr, aus Ihrem freundlichen Schreiben vom 20ten Oct. zu ersehen, daß jetzt Ihre Bedenken gegen das "Transfinitum" geschwunden sind. Gelegentlich will ich Ihnen aber einen kleinen Aufsatz verfaßen und zuschicken, in welchem ich in scholastischer Form detaillirt zeigen möchte, wie sich meine Resultate gegen die bekannten Argumente vertheidigen lassen und vor Allem, wie durch mein System die Grundlagen der christlichen Philosophie in allem Wesentlichen unverändert bleiben, nicht erschüttert, sondern vielmehr eher gefestigt werden, und wie sogar damit ihre Ausbildung nach verschiedenen Seiten gefördert werden kann. [Cantor an P. Ignatius Jeiler, 27. Okt. 1895]

Highly esteemed Pater Ign. Jeiler, I am very glad to see from your friendly letter of 20 Oct. that meanwhile your qualms against the "transfinitum" have disappeared. Betimes I will write and send you a little essay where I want to show you in scholastic form in detail how my results can be defended against the well-known arguments and, above all, how by my system the foundations of in all essentials remain unchanged, they are not shaken but rather become fixed, and how even their development in different directions can be promoted.

§ 361 Das Unendliche und die Theologie / Infinity and Theology (4)

Da ich nun aber auch keine Stütze von früheren wissenschaftlichen Autoritäten für meine Ansichten habe finden können, so weit ich auch in die beiden letzten Jahrhunderte, in das Mittelalter und selbst in das griechische Alterthum mich zurückversetzte, war es mir, so sonderbar es Ihnen vielleicht vorkommen wird, eine gewisse Befriedigung in Exodus, cap. XV, v. 18 wenigstens eine Art von Anklang an die transfiniten Zahlen zu finden, indem es dort heisst: "Dominus regnabit in infinitum (aeternum) et ultra". Ich meine dieses "et ultra" ist eine Andeutung dafür, dass es mit dem ω nicht sein Bewenden hat, sondern dass es auch darüber hinaus noch was giebt. [Cantor an Lipschitz, 19. Nov. 1883]

I have not been able to find support for my opinions from ancient scientific authorities, how far I went back into the two last centuries, into the Middle Ages and even into Greek antiquity. Therefore it was a certain satisfaction for me, how strange this may appear to you, to find in Exodus XV verse18 at least something reminiscent of transfinite numbers, namely the text: "God is king in eternity and beyond". I think this "and beyond" is a hint to the fact that ω is not the end but that something is existing beyond.

§ 362 Das Unendliche und die Theologie / Infinity and Theology (5)

Unter einem Actual Unendlichen ist dagegen ein Quantum zu verstehen, das einerseits nicht veränderlich, sondern vielmehr in allen seinen Teilen fest und bestimmt, eine richtige Konstante ist, zugleich aber andrerseits jede endliche Größe derselben Art an Größe übertrifft. Als Beispiel führe ich die Gesamtheit, den Inbegriff aller endlichen ganzen positiven Zahlen an; diese Menge ist ein Ding für sich und bildet, ganz abgesehen von der natürlichen Folge der dazu gehörigen Zahlen, ein in allen Teilen festes, bestimmtes Quantum, ein aphorismenon das offenbar größer zu nennen ist als jede endliche Anzahl. Fußnote: Man vgl. die hiermit übereinstimmende Auffassung der ganzen Zahlenreihe als aktual-unendliches Quantum bei S. Augustin (De civitate Dei. lib. XII, cap. 19): Contra eos, qui dicunt ea, quae infinita sunt, nec Dei posse scientia comprehendi. Wegen der großen Bedeutung, welche diese Stelle für meinen Standpunkt hat, will ich sie wörtlich hier aufnehmen [...] Indem nun der h. Augustin die totale, intuitive Perzeption der Menge (nü), "quodam ineffabili modo", a parte Dei behauptet, erkennt er zugleich diese Menge formaliter als ein aktual- unendliches Ganzes, als ein Transfinitum an, und wir sind gezwungen, ihm darin zu folgen. [Cantor an Prof. Dr. med. A. Eulenburg, Berlin, 28. Feb. 1886]

By the actual infinite we have to understand a quantity that is not variable but fixed and defined in all its parts, really a constant, but also exceeding every finite size of the same kind by size. As an example I mention the set of all finite positive integers; this set is a self-contained thing and forms, apart from the natural sequence of its numbers, a fixed and defined quantity, an aphorismenon, which is obviously larger than every finite number. Footnote: Compare the concurring perception of the whole sequence of number as an actually infinite quantum by S. Augustin (De civitate Dei. lib. XII, cap. 19): Contra eos, qui dicunt ea, quae infinita sunt, nec Dei posse scientia comprehendi. Because of its great importance for my position I will quote it here in full [...] While now the h. Augustin claims the total, intuitive perception of the set (nue), "quodam ineffabili modo", a parte Dei, he acknowledges this set formally as an actual infinite entity, as a transfinitum, and we are forced to follow him in this matter.

§ 363 Das Unendliche und die Theologie / Infinity and Theology (6)

Sie haben, soviel ich weiß, in Spanien 10 Universitäten, von denen aber die Theologie ausgeschloßen ist, welche bei Ihnen nur in Priesterseminaren gelehrt wird. Die frühere Einrichtung der Universitäten, wo die Theologie mit einbegriffen war, halte ich für die beßere, sowohl für Spanien, wie auch für Frankreich, wo ja derselbe Ausschluß eingeführt worden ist. Nicht nur lege ich Werth darauf, daß die übrigen Wissenschaften sich nicht principiell feindlich gegen die angestammte Theologie verhalten, sondern ich glaube, daß auch die Theologie durch eine engere Anlehnung an die übrigen Facultäten nur Nutzen für sich ziehen kann. [Cantor an Don Zoel Garcia de Gáldeano, 1893, zitiert in einem Brief von Cantor an Baumgartner, 15. Dez. 1893]

You have, as far as I know 10 universities in Spain. Theology however is excluded and is only taught in seminaries. The former constitution of universities, which included theology, is better in my opinion. That holds for Spain as well as for France, where the same exclusion has been introduced. I do not only care about a non-hostile attitude of the other sciences towards the ancestral theology, but I believe that also theology can only stand to gain from a close relation to the other faculties.

§ 364 Das Unendliche und die Theologie / Infinity and Theology (7)

Die Speculation, zumal die mathematische nimmt schon seit zehn Jahren ungefähr, nachdem ich die eigentlichen Hauptschwierigkeiten in Bezug auf das Transfinite überwunden hatte, nur einen geringen Theil meiner Zeit in Anspruch, die ich vielmehr der Theologie und "guten Werken" widme. [...] Sie können also sicher sein, daß ich an dem, Don Zoel Garcia de Galdeano gegenüber eingenommenen Standpuncte fest halten und, soweit es in meinen Kräften steht, das meinige dazu beitragen werde, in Spanien gesündere Zustände anzubahnen, wie ich dies auch für Italien und Frankreich durch meine Verbindungen mit den dortigen Mathematikern seit vielen Jahren erstrebe. [...] Zum Begriffe einer Universität gehört das einigermaaßen friedliche Zusammenleben und Wirken der vier Facultäten; und wenn dieses Verhältniß seit der Reformation in vielen catholischen Ländern zuerst in‘s Schwanken gekommen und nachher ganz aufgelöst worden ist, so muß Alles geschehen (mit Vorsicht und Klugheit selbstverständlich) um jenen, allein naturgemäßen Zustand mit der Zeit wieder herbeizuführen. [Cantor an P. Alexander Baumgartner SJ, 27. Dez. 1893]

The speculation, in particular the mathematical one, occupies only a small part of my time, after having overcome the original main difficulties with respect to the tranfinite. I devote the speculation to theology and "good works". [...] You can be sure that I will adhere to the standpoint, mentioned to Don Zoel Garcia de Galdeano, and contribute according to my power to initiate healthier states in Spain, as I have been trying for years for Italy and France by means of my relations with the mathematicians there. [...] The institution university requires the rather peaceful collaboration of the four faculties. Caused by the reformation this relation in many catholic countries first has been shaken and then completely deleted. Everything has to be done (with care and cleverness, of course) to reestablish step by step this only natural state.

§ 365 Das Unendliche und die Theologie / Infinity and Theology (8)

Sie würden auch ein "gutes Werk“ thun, wenn Sie darauf hinwirken wollten, daß gelegentlich, wenn auch nur auf kurze Zeit, einige Ihrer jüngeren, der Metaphysik sich widmenden Patres hierher zu mir geschickt würden, um mit mir über das actuale Unendliche, (diese „quaestio multis molestißima de infinita multitudine“ wie sich Card. Franzelin in seinem Tr. de Deo uno sec. nat. Thes. XLI ausdrückt) privatißime zu disputiren. Denn Sie können sich darauf verlaßen, daß, der Standpunct, den in dieser Frage die Mehrzahl Ihrer Patres (aber auch die Mehrzahl der kathol. Theologen) einnimmt, auf die Dauer völlig unhaltbar ist. [...] bemerke ich, daß größere Vorkenntniße in der Mathematik zum Verständniß meiner Lehre nicht nöthig sind, sondern nur eine gründliche philosophische Vorbildung, wie sie ja bei Ihnen auf‘s Beste und Schönste erlangt wird. [...] Das einzige, wofür ich diesem Langbehn {{deutscher Erfolgsautor des Buches "Rembrandt"}} dankbar bin, ist, daß er [...] mich auf eine angebliche Aehnlichkeit meines Kopfes und Gesichts mit dem heiligen Ignatius von Loyola aufmerksam machte. Deßen Exercitien kenne und lese ich seit vielen Jahren. Möglicherweise ist aber auch dieser Vergleich ein ebensolcher Quatsch und ebenso verrückt, wie die meisten Vergleiche seines "Rembrandt“. [...] Soll übrigens meine Wirksamkeit in Spanien erfolgreich werden, so würde ich für eine geschickte Cooperation Ihrer Patres in verwandtem Sinne sehr dankbar sein. [Cantor an P. Alexander Baumgartner SJ, 27. Dez. 1893]

You would also do a "good work" if you would push some of your younger, metaphysically interested patres to occasionally visit me for a short time and discuss privatissime with me about the actual infinite, (this "quaestio multis molestissima de infinita multitudine“ as Card. Franzelin calls it in his Tr. de Deo uno sec. nat. Thes. XLI). You can be sure that the point of view of the majority of patres (but also of the catholic theologians) is in the long term completely untainable. [...] I mention that great previous knowledge of mathematics is not required to understand my techings, but only extensive philosophical knowledge, as it is learned best and most beautifully at your institution. [...] The only reason I have to be grateful to Langbehn {{German author of the best seller "Rembrandt"}} is his hint to his observation that my head allegedly resembles the face of the holy Ignatius of Loyola. I know his spiritual exercises and have been reading them for many years. Perhaps this has had an influence on my looks. But perhaps this comparison is as much nonsense and silly as most comparisons in his "Rembrandt". [...] Should my agitation in Spain become successful, I would be very grateful for a clever cooperation of your patres in kindred spirit.

§ 366 Das Unendliche und die Theologie / Infinity and Theology (9)

Was die dritte auf das A. U bezügliche Frage, nämlich nach dem A. U in Deo aeterno omnipotenti seu in natura naturante betrifft, (den letzteren Ausdruck habe ich einigen grossen Scholastikern entnommen) so zweifle ich nicht, dass wir hier wieder hinsichtlich der Bejahung ganz einer Ansicht sind. Das letztere A.U, d.h. das A.U in Deo, nenne ich, wie Sie in meinem Schriftchen "Grundlagen" bemerkt haben werden, das Absolute und es fällt dasselbe ganz ausserhalb der Zahlentheorie. Dagegen sind das A. U in abstracto und in concreto, wo ich es Transfinitum nenne, nicht nur Gegenstand einer erweiterten Zahlentheorie, sondern auch, wie ich noch zu zeigen hoffe, einer avancirten Naturwissenschaft und Physik. [Cantor an P. Ignace Carbonnelle SJ, 28. Nov. 1885, aus C. Tapp: "Kardinalität und Kardinäle", Franz Steiner Verlag (2005)]

With respect to the third question concerning the A. I {{actual infinite}}, namely the A. I in Deo aeterno omnipotenti seu in natura naturante (the last expression I have adopted from some great scholastics) I have no doubt that we agree again in its approval. The last A. I, i.e., the A. I in Deo, I call the Absolute, as you will have noted in my little essay "Grundlagen", and this falls completely out of number theory. The A. I in abstracto and in concreto, however, where I call it transfinitum, are not only subject of an extended number theory but also, as I hope to show, of an advanced natural science and physics.

§ 367 Das Unendliche und die Theologie / Infinity and Theology (10)

Mit großem Interesse habe ich Ihre Schrift: "Die Lehre des hl. Thomas von Aquino über die Möglichkeit einer anfangslosen Schöpfung“ studiert. Es war mir eine innige Befriedigung, von so berufener Seite die Stellung des heil. Thomas zur Frage des actualen Unendlichen erörtert zu sehen und mich zu überzeugen, daß ich den heil. Thomas in diesem Puncte und den damit zusammenhängenden Fragen richtig verstanden habe, daß vor Allem seine Argumentation gegen das actuale Unendliche in creatis resp, gegen die Möglichkeit act. unendl. großer Zahlen für ihn selbst nicht die Bedeutung einer demonstratio, quae usquequaque de necessitate concludit und metaphysische Gewissheit liefert, gehabt hat; sondern sie war in seinen eigenen Augen nur in gewissem Grade probabel. [...] Ich stehe überall auf demselben Boden wie Sie und freue mich daher umsomehr, daß aus Ihrem Schreiben Ihre Absicht hervorzugehen scheint, meine Lehre vom Transfiniten einer gründlichen Prüfung zu unterwerfen. [Cantor an P. Thomas Esser 0P, 5. Dez. 1895, aus C. Tapp: "Kardinalität und Kardinäle", Franz Steiner Verlag (2005)]

With great interest I have studied your essay: The teachings of holy Thomas of Aquino about the possibility of a creation without beginning. It was very satisfying for me to see the position of holy Thomas concerning actual infinity be discussed from such a profound expert and to learn that I had correctly understood holy Thomas in this point and related questions, in particular that his arguments against the actual infinite in creatis or against the possibility of actually infinitely great numbers has, for himself, not the meaning of a demonstratio, quae usquequaque de necessitate concludit leading to metaphysical certainty, but was in his own eyes only probably in a certain degree. [...] I stand everywhere on the same ground as you and I am the more happy that your letter appears to contain your intention to thoroughly examine my teachings of the transfinite.

§ 368 Das Unendliche und die Theologie / Infinity and Theology (11)

Die allgemeine Mengenlehre [...] gehört durchaus zur Metaphysik. Sie überzeugen sich hiervon leicht, wenn sie die Kategorieen der Kardinalzahl und des Ordnungstypus, dieser Grundbegriffe der Mengenlehre, auf den Grad ihrer Allgemeinheit prüfen. [...] und auch der Umstand, dass die unter meiner Feder noch stehende Arbeit in mathematischen Journalen herausgegeben wird, modificirt nicht den metaphysischen Inhalt und Charakter derselben. [...] Von mir wird der christlichen Philosophie zum ersten Mal die wahre Lehre vom Unendlichen in ihren Anfängen dargeboten. Ich weiß ganz sicher und bestimmt, dass sie diese Lehre annehmen wird, es fragt sich nur, ob schon jetzt oder erst nach meinem Tode. Dieser Alternative stehe ich vollkommen gleichmüthig gegenüber, sie berührt nicht meine arme Seele, die ich vielmehr, verehrter Pater, in Ihr und der Ihrigen frommes Gebet empfehle. [Cantor an P. Thomas Esser OP, 15. Feb. 1896, aus C. Tapp: "Kardinalität und Kardinäle", Franz Steiner Verlag (2005)]

The general set theory [...] definitely belongs to metaphysics. You can easily convince yourself when examining the categories of cardinal numbers and the order type, these basic notions of set theory, on the degree of their generality. [...] and the fact that my presently written work is issued in mathematical journals does not modify the metaphysical contents of this work. [...] By me christian philosophy is for the first time confronted with the true teachings of the infinite in its beginnings. I know quite firmly and certainly, that my teachings will be accepted. The question is only, whether this will happen before or after my death. But I am completely calm about this alternative. It does no touch my poor soul wich, however, dear Pater, I recommend to your and yours pious prayer.

§ 369 Das Unendliche und die Theologie / Infinity and Theology (12)

Am Meisten würde es mich aber freuen, wenn meine Arbeiten {{über transfinite Cardinalzahlen und transfinite Ordnungstypen}} der meinem Herzen am Nächsten stehenden christlichen Philosophie, der "philosophia perennis“, zugute kämen, was nur dann denkbar und möglich wäre, wenn sie von der alten, nun durch S. Heiligkeit Leo XIII. so herrlich erneuerten, wiedererstandenen Schule genau und eingehend untersucht und geprüft würden. [Cantor an R. P. Thomas Esser 0P, 19. Dez. 1895]

I would be most happy if my works {{on transfinite cardinal numbers and transfinite order types}} would be for the benefit of the christian philosophy which is next to my heart, namely the "philosophia perennis". This would onyl then be thinkable and possible, if they would be scrutinized by the old, meanwhile by His Holiness Leo XIII so beautifully restored, revived school.

§ 370 Das Unendliche und die Theologie / Infinity and Theology (13)

Ein Wesen existiert, das alle positiven Eigenschaften in sich vereint. Das bewies der legendäre Mathematiker Kurt Gödel mit einem komplizierten Formelgebilde. Zwei Wissenschaftler haben diesen Gottesbeweis nun überprüft - und für gültig befunden. [...] Die Existenz Gottes kann fortan als gesichertes logisches Theorem gelten. {{So wie viele Sätze in ZFC. Dies ist zweifellos ein wichtiges Beispiel für die durch automatische Beweisprüfer gewonnene Sicherheit.}} http://www.spiegel.de/wissenschaft/mensch/formel-von-kurt-goedel-mathematiker-bestaetigen- gottesbeweis-a-920455.html

A being exist which reconciles all positive properties in itself. That has been proven by the legendary mathematician Kurt Goedel by means of a complicated formula. Two scientists have scrutinized this proof of God - and have approved it. [...] The existence of God can in future be assumed to be a proven logical theorem. {{Like many theorems of ZFC.}} [Translated from the German text of SPIEGEL-ONLINE http://www.spiegel.de/wissenschaft/mensch/formel-von-kurt-goedel-mathematiker-bestaetigen- gottesbeweis-a-920455.html

Goedel's ontological proof has been analysed for the first-time with an unprecedent degree of detail and formality with the help of higher-order theorem provers. {{An important example for the advantage of formalizing and the safety gained by checking theorems by means of theorem provers}} Christoph Benzmüller, Bruno Woltzenlogel Paleo: "Formalization, Mechanization and Automation of Gödel's Proof of God's Existence", Arxiv (2013) http://arxiv.org/abs/1308.4526

§ 371 Das Unendliche und die Theologie / Infinity and Theology (14)

Gestatten Sie, Monsignore, dass ich Ihnen beifolgend einen kleinen Aufsatz (in Correctur) überreiche, von dem ich mir erlauben werde, Ihnen einige Exemplare unter Kreuzband zu senden, sobald die Abzüge vollendet sein werden. Es würde mich freuen, wenn der darin enthaltene Versuch, die drei Hauptfragen mit Bezug auf das actuale Unendliche gehörig abzugrenzen, einer Prüfung auch seitens der christlich- catholischen Philosophen unterzogen würde. [Cantor an S. Eminenz Cardinal Joannis Baptistae Franzelin SJ, 17. Dez. 1885]

Ich kann mich gegenwärtig mit metaphysischen Erörterungen wenig beschäftigen; gestehe jedoch, dass nach meiner Meinung das, was der Herr Verfasser das „Transfinitum in natura naturata“ nennt, sich nicht vertheidigen lässt, und in einem gewissen Sinne, den ihm der Herr Verfasser jedoch nicht zu geben scheint, den Irrthum des Pantheismus enthalten würde. [Antwort von Franzelin an Cantor, 25. Dez. 1885]

Versuche, die ich schon vor vielen Jahren und neuerdings wiederholt gemacht habe, Mitglieder der deutschen Provinz der S. J. zu einer solchen vertraulichen wissenschaftlichen Correspondenz über das Actual-unendliche zu veranlassen, sind, obgleich viele von ihnen meine Arbeiten seit mindestens zehn Jahren kennen und in Händen haben, ohne jeden Erfolg geblieben, während doch der hochselige Cardinal J. B. Franzelin in seinen gerade vor 10 Jahren an mich gerichteten Briefen auf die Bedeutung der Frage für Theologie u. Philosophie deutlich genug hingewiesen hat. [Cantor an R. P. Thomas Esser 0P, 19. Dez. 1895]

Monsignore, may I present you the included galley proofs of the little essay, of which I will send you some copies as soon as it has been completed. I would be glad if my attempt to properly distinguish between the three main questions with respect to the actual infinite could be scrutinized thoroughly by christian-catholic philosophers. [Cantor to Cardinal Joannis Baptistae Franzelin SJ, 17 Dec. 1885]

Presently I am rather unable to consider metaphysical arguments. But I confess that in my opinion that which is called by the author the "Transfinitum in natura naturata“, cannot be defended and in a certain sense, which however the author does not seem to claim, would include the error of pantheism. [Answer from Franzelin to Cantor, 25 Dec. 1885]

Attempts that I have made many years ago and repeatedly recently, to win members of the German province of S. J. {{the Jesuites}} for a confidential scientific correspondence about the actual infinite, have been without success although many of them have been knowing and possessing my works for more than ten years, whereas the late Cardinal J.B. Franzelin very plainly has been pointing to the importance of this question for theology and philosophy in his letters directed to me. [Cantor to R. P. Thomas Esser 0P, 19 Dec. 1895]

§ 372 Das Unendliche und die Theologie / Infinity and Theology (15)

Dementsprechend unterscheide ich ein "Infinitum aeternum sive Absolutum", das sich auf Gott und seine Attribute bezieht, und ein "Infinitum creatum sive Transfinitum", das überall da ausgesagt wird, wo in der Natura creata ein Actualunendliches constatirt werden muss, wie beispielsweise in Beziehung auf die, meiner festen Ueberzeugung nach actual unendliche Zahl der geschaffenen Einzelwesen, sowohl im Weltall, wie auch schon auf unsrer Erde und, aller Wahrscheinlichkeit nach, selbst in jedem noch so kleinen ausgedehnten Theil des Raumes, worin ich mit Leibniz ganz übereinstimme. [Cantor an S. Eminenz Cardinal Joannis Baptistae Franzelin SJ, 22. Jan. 1886]

Accordingly I distinguish an "Infinitum aeternum sive Absolutum" that refers to God and his attributes, and an "Infinitum creatum sive Transfinitum" that has to be applied wherever in the Natura creata an actual infinite is observed, like, for example, with respect to the, according to my firm conviction, actually infinite number of created individuals, in the universe as already on our earth and, most probably, even in each extended part of the space, however small it may be. Here I agree completely with Leibniz.

§ 373 Das Unendliche und die Theologie / Infinity and Theology (16)

Obwohl ich weiss, dass diese Lehre vom "Infinitum creatum", wenn auch nicht von allen, doch von den meisten Kirchenlehrern bekämpft wird und im Besonderen auch vom grossen St Thomas Aquinatus [...] so sind doch die Gründe, welche in dieser Frage im Verlauf zwanzigjähriger Forschung [...] sich mir aufgedrängt und mich gewissermaaßen gefangen genommen haben, stärker, als Alles, was ich bisher dagegen gesagt fand, obgleich ich es in weitem Umfange geprüft habe. Auch glaube ich, dass die Worte der heiligen Schrift, wie z. B. Sap. c. 11, v. 21: "Omnia in pondere, numero et mensura disposuisti" in denen ein Widerspruch gegen die actual unendlichen Zahlen vermuthet wurde, diesen Sinn nicht haben; denn gesetzt den Fall, es gäbe, wie ich bewiesen zu haben glaube, actual unendliche "Mächtigkeiten“ d. h. Cardinalzahlen und actualunendliche "Anzahlen wohlgeordneter Mengen“ d. h. Ordinalzahlen [...] die ebenso wie die endlichen Zahlen feste, von Gott gegebene Gesetze befolgen, so würden ganz sicherlich auch diese transfiniten Zahlen in jenem heiligen Ausspruche mitgemeint sein und es darf daher, meines Erachtens, derselbe nicht als Argument gegen die actual unendlichen Zahlen genommen werden, wenn ein Cirkelschluss vermieden werden soll. Dass aber ein „Infinitum creatum“ als existent angenommen werden muß, läßt sich mehrfach beweisen. [...] Ein Beweis geht vom Gottesbegriff aus und schliesst zunächst aus der höchsten Vollkommenheit Gottes Wesens auf die Möglichkeit der Schöpfung eines Transfinitum ordinatum, sodann aus seiner Allgüte und Herrlichkeit auf die Nothwendigkeit einer thatsächlich erfolgten Schöpfung des Transfinitum. Ein andrer Beweis zeigt a posteriori, dass die Annahme eines "Transfinitum in natura naturata“ eine bessere, weil vollkommenere Erklärung der Phänomene, im Besondern der Organismen und psychischen Erscheinungen ermöglicht, als die entgegengesetzte Hypothese. {{Letzteres wurde von Cantor niemals weiter ausgeführt.}} [Cantor an S. Eminenz Cardinal Joannis Baptistae Franzelin SJ, 22. Jan. 1886]

Although I know that this teaching of the "Infinitum creatum", is objected, if not by all, yet by most doctors of the church, and in particular by the great St Thomas Aquinatus [...] the reasons that have imposed themselves on me and rather captivated me during 20 years of reasearch [...] are stronger than everything contrary I have heard, although I have checked that very carefully. Further I believe that the words of the Holy Bible like Sap. c. 11, v. 21: "Omnia in pondere, numero et mensura disposuisti“ which have been assumed to contradict infinite numbers, do not have that meaning. Given the case, actually infinite "powers", i.e., cardinal numbers and actually infinite "numbers of well-ordered sets", i.e., ordinal numbers [...] existed, as I think to have proved, which like finite numbers obey firm laws given by God, so clearly also these transfinite numbers would be covered by that holy remark - and it cannot be used against actually infinite numbers if a circular argument shall be avoided. It can be proved in different ways that an "Infinitum creatum“ has to be assumed. [...] One of the proofs starts from the notion of God and concludes first from the highest perfection of the Supreme Being on the possibility of the creation of a Transfinitum ordinatum. Then from God's loving kindness and glory on the necessity of an actually created Transfinitum. Another proof shows a posteriori that the assumption of a "Transfinitum in natura naturata“ delivers a better, more complete, explanation of the phenomena, in particular of the organisms and physical phenomena than the contrary hypothesis. {{This has never been further elaborated by Cantor.}}

§ 374 Das Unendliche und die Theologie / Infinity and Theology (17)

Vom Pantheismus glaube ich jedoch, dass er, und vielleicht nur durch meine Auffassung der Dinge, mit der Zeit ganz überwunden werden könnte. [...] Was aber den Materialismus und die damit zusammenhängenden Richtungen betrifft, so scheinen sie mir, gerade weil sie die wissenschaftlich unhaltbarsten und am Leichtesten widerlegbaren sind, zu den Uebeln zu gehören, von welchen das menschliche Geschlecht in dem zeitlichen Dasein nie ganz zu befreien sein wird. [Cantor an S. Eminenz Cardinal Joannis Baptistae Franzelin SJ, 22. Jan. 1886]

In Ihrem werthen Schreiben an mich sagen Sie nämlich erstens ganz richtig (vorausgesetzt daß Ihr Begriff des Transfinitum nicht blos religiös unverfänglich, sondern auch wahr ist, worüber ich nicht urtheile), "ein Beweis geht vom Gottesbegriffe aus und schließt zunächst aus der höchsten Vollkommenheit Gottes Wesens auf die Möglichkeit der Schöpfung eines Transfinitum ordinatum.“ In der Voraussetzung, daß Ihr Transfinitum actuale in sich keinen Widerspruch enthält, ist Ihr Schluß auf die Möglichkeit der Schöpfung eines Transfinitum aus dem Begriffe von Gottes Allmacht ganz richtig. Allein zu meinem Bedauren gehen Sie weiter und schließen "aus seiner Allgüte und Herrlichkeit auf die Nothwendigkeit einer thatsächlich erfolgten Schöpfung des Transfinitum". Gerade weil Gott an sich das absolute unendliche Gut und die absolute Herrlichkeit ist, welchem Gute und welcher Herrlichkeit nichts zuwachsen und nichts abgehen kann, ist die Nothwendigkeit einer Schöpfung, welche immer diese sein mag, ein Widerspruch. [Franzelin an Cantor, 26. Jan. 1886]

I believe that pantheism, perhaps only by means of my theory of the things, can be overcome completely. [...] and related ideas seem to me to belong to the evils of which, just because they belong to the scientifically most untenable and easiest refutable, the human race in its temporal existence will never be completely be released of. [Cantor to Cardinal Joannis Baptistae Franzelin SJ, 22 Jan. 1886]

In your valued letter to me you say first quite right (provided that your notion of the transfinitum is not only compatible with religion but also true, what I do not judge), "one of the proofs starts from the notion of God and concludes first from the highest perfection of the Supreme Being on the possibility of the creation of a transfinitum ordinatum." Assuming that your transfinitum actuale is free of contradictions your conclusion on the possibility of the creation of a transfinitum out of the notion of God's omnipotence is quite right. But to my regret you go on and conclude from his "loving kindness and glory on the necessity of an actually created transfinitum". Just because God himself is the absolute infinite good and the absolute glory, which good and which glory nothing can be added and nothing can be missing, the necessity of some creation, whatever it might be, is a contradiction. [Franzelin to Cantor, 26 Jan. 1886]

§ 375 Das Unendliche und die Theologie / Infinity and Theology (18)

Auf eine weitere Korrespondenz über Ihre philosophischen Ansichten kann ich bei meinen vielen Beschäftigungen, durch welche ich auf ein ganz anderes Feld angewiesen bin, mich ferner nicht einlaßen; Sie mögen mich also entschuldigen, wenn ich auf Ihre etwaigen Repliken, welche ich jedoch insoweit sie sich auf Ihr System beziehen zu unterlaßen bitte, nicht werde antworten können. [Franzelin an Cantor, 26. Jan. 1886]

{{Wie jeder von seiner Sache Überzeugte konnte es aber auch Cantor nicht unterlassen, zu antworten. Er richtete noch zwei Briefe an den Kardinal, nämlich am 29. Jan. 1886 und in anderem Zusammenhang am 18. Feb. 1886, worauf jedoch keine Antwort erfolgte. Eine vollständige Sammlung aller bekannten Korrespondenz Cantors im geistlichen Umfeld findet sich in C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005).}}

Ewr. Eminenz ich meinen herzlichsten Dank für die Ausführungen des gütigen Schreibens vom 26ten dss., denen ich mit voller Ueberzeugung zustimme; denn in der kurzen Andeutung meines Briefes v. 22. dss. war es an der betreffenden Stelle nicht meine Meinung, von einer objectiven, metaphysischen Nothwendigkeit zum Schöpfungsact, welcher Gott, der absolut Freie unterworfen gewesen wäre, zu sprechen, sondern ich wollte nur auf eine gewisse subjective Notwendigkeit für uns hindeuten, aus Gottes Allgüte und Herrlichkeit auf eine thatsächlich erfolgte (nicht a parte Dei zu erfolgende) Schöpfung, nicht bloss eines Finitum ordinatum, sondern auch eines Transfinitum ordinatum zu folgern. [Cantor an S. Eminenz Cardinal Joannis Baptistae Franzelin SJ, 29. Jan. 1886]

I am not able to continue the correspondence about your philosophical opinions with you because of my many occupations which direct me to quite another field. You might excuse if I will not react on your possible replies, which however, as far as they will be related to your system, I beg you to refrain from. [Franzelin to Cantor, 26 Jan. 1886]

{{Like everyone who is sure of his ground Cantor could not refrain from answering. He directed two further letters to the Cardinal, on 29 Jan. 1886 and, with another topic, on 18 Feb. 1886, which however remained without reply. A complete collection of Cantor's known correspondence with clerics is supplied by C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005).}}

Your Eminence, I thank you very much indeed for the clarification given in your kind letter of 26 January which I agree to with full conviction, because in the short hint in my letter of 22 january I did not opine to talk about an objective, of the act of creation, which God, the absolutely Free had been subject to, but I only wanted to point to a certain subjective necessity for us, to conclude from God's loving kindness and glory on an actually done (not a parte Dei to be done) creation, not only of a Finitum ordinatum but also of a Transfinitum ordinatum. [Cantor to Cardinal Joannis Baptistae Franzelin SJ, 29 Jan. 1886]

§ 376 Here are some arguments of the orthodox internet matheologian Hancher alias Virgil, standing in place of many others, concerning the matrices of FISONs (Finite Initial Sequences Of Naturals)

1 1, 2 1, 2, 3 ... or 1 2, 1 3, 2, 1 ... that cannot have more lines than columns.

He said:

- no row ends without a last n and no column starts without a first n, so where does an unpaired row or column fit into that diagram? - there must be precisely the same number of rows as columns - the completion of that diagram has exactly as many rows as columns - Nor more columns than lines, whether truncated finitely or carried to infinite completion. - There are precisely the same number of rows as columns, and both are the same as the number of naturals. - So how can the number of rows differ from the number of columns if they both are in one-to- one correspondence with the set of naturals? - No one has opposed that no FISON has ¡0 elements. - there are ¡0 different FISONs

At least one of these statements is wrong, namely in contradiction with the others.

§ 377 Das Unendliche und die Theologie / Infinity and Theology (19)

[...] weit gewichtigere Gründe hinzufügen lassen, die aus der absoluten Omnipotenz Gottes fließen und denen gegenüber jede Negation der Möglichkeit eines "Transfinitum seu Infinitum actuale creatum" wie eine Verletzung jenes Attributes der Gottheit erscheint. [Cantor an Prof. Dr. Constantin Gutberlet, 24. Jan. 1886]

[...] far more Important reasons can be added which result from the absolute omnipotence of God and with respect to which every negation of the possibility of a "Transfinitum seu Infinitum actual creatum" appears like a violation of that attribute of God.

§ 378 Das Unendliche und die Theologie / Infinity and Theology (20)

Soll ich auch einen Punct erwähnen, worin ich mit Ihnen nicht ganz einverstanden bin, so ist es das unbedingte Vertrauen, welches von Ihnen dem modernen sogenannten Gesetz von der Erhaltung der Energie entgegengebracht wird. Ich will durchaus nicht die Lehre von der Aequivalenz der verschiedenen sich ineinander umsetzenden natürlichen Kraftformen in Zweifel ziehen, soweit sie experimentell hinreichend begründet ist. Das wogegen ich ernste Bedenken hege, ist sowohl die Erhebung des angeblichen Gesetzes zu einem metaphysischen Prinzip, von dem die Erkenntniss so gewichtiger Sätze, wie die Unsterblichkeit der Seele abhängig sein soll wie auch die von den Herren Thomson, v. Helmholtz, Clausius und Genossen beliebte und durch nichts gerechtfertigte Ausdehnung und Anwendung des Satzes von der Erhaltung der Energie auf das Weltganze, woran phantastische Speculationen geknüpft werden, die ich für ganz werthlos halte. {{Tja, wie man sich (und andere) täuschen kann.}} [Cantor an Prof. Dr. Constantin Gutberlet, 1. Mai 1888]

Should I also mention a point where I do not quite agree with you, so it is your unreserved confidence in the modern so-called law of energy conservation. I do not wish to doubt the teaching of the equivalence of the different natural forces transforming into each other as far as this has been experimentally verified. That wich I have serious reservations against is the elevation of the asserted law into the rank of a metaphysical priciple which governs the recognition of so important theorems as the immortality of the soul as well as its completely unjustified extension and application onto the whole world system as the gentlemen Thomson, v. Helmholtz, Clausius, and comrades, like to do, who add phantastic speculations which in my opinion are without any value. {{Amazing, how ignorant one can be (and make others)}}

§ 379 Das Unendliche und die Theologie / Infinity and Theology (21)

Der Aufsatz von Pohle über die objective Bedeutung des unendlich Kleinen enthält sehr schöne und gehaltreiche Betrachtungen. Nur irrt er mit der Annahme, daß das unendlich Kleine als actuelles integrirendes oder constitutives Element zur Erklärung des Continuums resp. zur Begründung der Infinitesimalrechnung nothwendig sei. Ich trete mit ihm für die objective Bedeutung des unendliche Kleinen ein, doch nicht des unendlich Kleinen, sofern es ein actuelles, unendlich klein seiendes wäre, vielmehr nur sofern es ein potenzielles, unendlich klein werdendes ist. Als Element des Continuums ist das Unendlichkleine nicht bloß unbrauchbar, sondern auch an sich undenkbar resp. unmöglich, wie ich streng beweisen kann. [Cantor an Prof. Dr. Constantin Gutberlet, 1. Mai 1888]

The paper by Pohle about the objective meaning of the infinitely small contains quite nice and comprehensive reflections. But he errs in the assumption that the infinitely small be necessary as an actually integrating or constituent element for the explanation of the continuum or as a foundation of the infinitesimal calculus. I agree with him concerning the objective importance of the infinitely small, but not the infinitely small as far as it is something actual, being infinitely small, rather only something potential, becoming infinitely small. As an element of the continuum the infinitely small is not only unusable but even unthinkable or impossible as I can strictly prove.

§ 380 Resolution of the dissent between mathematics and matheology. Hancher alias Virgil, one of the defenders of matheology, explains the difference:

WM: How can you dare to say that ¡0 columns are spanned by FISONs in the table

1 1, 2 1, 2, 3 ...

Virgil: Because I dare speak truth.

WM: If not every natural is in one and the same FISON, then at least two FISONs are required to contain every natural - two or more or infinitely many. But everybody who claims that this is true should be able to name the first FISON of the required set of FISONs.

Virgil: If that were so, then somewhere in wikipedia or elsewhere on the net somebody other than WM must have posted that "fundamental law of arithmetic".

WM: Every set of FISONs has a first element.

Virgil: That would only apply if there were only one set of FISONs whose union was Ù.

From these few sentences everybody can recognize the basic requirements of matheology - in addition to the fact that enumerating infinite sets is a super task (cp. § 347) that cannot be accepted as part of mathematics and that the resulting idea of uncountable sets implies the "existence" of undefinable "real" numbers.

§ 381 Das Unendliche und die Theologie / Infinity and Theology (22)

Da er {{Cantor}} sich wegen dieses kühnen Unternehmens von allen Seiten angegriffen sah, suchte er Sukkurs bei mir, dem einzigen, der, wie er glaubte, mit seiner Auffassung übereinstimmte. Da er von edler Gesinnung war, teilte er nicht die Verachtung, mit welcher die ungläubige Wissenschaft die christlichen Philosophen behandelt. Es war auch nicht die bloße Not, welche ihn zu mir führte, sondern, wie er sagte, habe er darum eine katholikenfreundliche Gesinnung, weil seine Mutter katholisch war. Er befragte mich über die Lehre der Scholastiker in betreff dieser Frage. Ich konnte ihn besonders auf den hl. Augustin und auf den P. Franzelin, den späteren Kardinal, hinweisen. Dieser mein hochverehrter Lehrer verteidigte die aktual unendliche Menge in der Erkenntnis Gottes, gestützt auf die ausdrückliche Lehre des hl. Augustin, und er war es, der mir den Anstoß zu jener Schrift gegeben, und mich bei den heftigen Angriffen damit beruhigte, daß ich nur die Lehre des hl. Augustin vortrage. An den Kardinal wandte sich Cantor selbst, und Äußerungen desselben teilt er, ohne ihn zu nennen, in einem Aufsatze der "Zeitschrift für Philosophie und philosophische Kritik" mit. [C. Gutberlet: Philos. Jahrbuch der Görres-Ges. 32 (1919) 364]

Since he {{Cantor}}, because of his bold endeavor, had been attacked from all sides, he tried to get support from me, the only one who, as he believed, agreed with his opinions. Since he was noble minded, he did not share the contempt of the disbelieving science against the christian philosophers. And it was not only pure poverty which lead him to me, but, as he said, he had a catholic-friendly attidude because his mother was catholic. He inquired with me about the teachings of the scholastics with respect to this question. I could point him in particular to St. Augustin and to P. Franzelin, the later cardinal. This highly esteemed teacher of mine defended the actually infinite set in the cognition of God, supported by the explicit teaching of St. Augustin, and it has been he, who induced that writing of mine and who calmed me during the violent attacks with the argument that I only had repeated the teaching of St. Augustin. Cantor himself then addressed the cardinal and reported his statements, without revealing his name, in an essay of the "Zeitschrift für Philosophie und philosophische Kritik".

§ 382 Das Unendliche und die Theologie / Infinity and Theology (23)

Ich bin niemals von einem "Genus supremum" des actualen Unendlichen ausgegangen. Ganz im Gegentheil habe ich streng bewiesen, daß es ein "Genus supremum" des actualen Unendlichen garnicht giebt. Was über allem Finiten und Transfiniten liegt, ist kein "Genus"; es ist die einzige, völlig individuelle Einheit, in der Alles ist, die Alles umfasst, das "Absolute", für den menschlichen Verstand Unfassbare, also der Mathematik gar nicht unterworfene, Unmessbare, das "ens simplicissimum", der "Actus purissimus", der von Vielen "Gott" genannt wird. [Cantor an Mrs. Chisholm-Young, 20. Juni 1908]

I have never been assuming a "Genus supremum" of the actual infinite. On the contrary I have proven strictly that a "Genus supremum" of the actual infinite does not exist. That which is higher than all finite and transfinite is not a "Genus", it is the only absolutely individual unit, which contains all, which comprehends all, the "Absolute", for the human intellect incomprehensible, therefore not being subject to mathematics, unmeasurable, the "ens simplicissimum", the Actus purissimus, which by many is called "God".

§ 383 Das Unendliche und die Theologie (24)

Nun hat Herr Bernstein die neue Unvorsichtigkeit begangen, in den mathem. Annalen zeigen zu wollen, daß es "Mengen giebt, die nicht wohlgeordnet werden können". Ich habe keine Zeit nach dem Fehler in seinem Beweise zu suchen, bin aber fest überzeugt, daß ein solcher vorhanden ist. Hoffentlich kommt bald die Zeit und Gelegenheit, wo ich meine volle Meinung über alle derartigen prämaturirten Versuche aussprechen kann. [...] Das Fundament für meine Auffassung des Erlösungswerkes ist, daß Jesus der vorausgesagte Messias der Juden und als solcher seiner Menschheit nach ein richtiger Nachkomme Davids ist. Dies wissen wir auf's Sicherste von ihm selbst und als solcher gilt er nach seiner Auferstehung allen seinen Aposteln. Von hier aus komme ich, wie Sie gesehen haben, auf Grund des neuen Testaments dazu, zwei Josephs zu unterscheiden, den Königl. Joseph und leiblichen Vater Christi und den Nährvater Joseph. [...] Was die Auferstehung Christi betrifft (zu welcher Sie meine Stellung wissen wollen), so ist sie durch die Schriften des neuen Testaments aufs Beste und Umfassendste bezeugt; ich glaube fest daran, als an eine Thatsache und grüble nicht über das "Wie" derselben. [Cantor an Jourdain, 3. Mai 1905]

Recently Mr. Bernstein has committed the new carelessness, to try to show in the mathem. Annalen that "there are sets existing which cannot be well-ordered". I have not the time to look for the error in his proof but I am firmly convinced that such an error exists. Hopefully time and opportunity will come soon to frankly express my full opinion about all those immature attempts. [...] The fundament of my opinion about redemption is that Jesus is the predicted Messiah of the Jews and as such in his human nature a real descendant of David. This we know absolutely sure from himself and as such he has been considered by all his apostles after his resurrection. From this point I arrive, as you have seen, based on the New Testament, at the distinction of two Josephs, the royal Joseph and physical father of Christ and the breadwinner Joseph. [...] Concerning the resurrection of Christ (about which you have inquired me), this has been attested best and most comprehensively by the writings of the New Testament. I firmly believe it as a fact and do not brood over the "how" of it.

§ 384 Das Unendliche und die Theologie / Infinity and Theology (25)

Wie mir ein mir persönlich bekannter Ordensbruder des Autors erzählte, ist P. Esser momentan in Rom als Mitarbeiter an einer hochwichtigen vom heil. Vater berufenen Commißion zur Revision des Index [...] {{Diese "hochwichtige" Kommission erneuerte den Index der verbotenen Bücher, d.h. der Bücher, die ein gläubiger Katholik nicht lesen durfte.}} [Cantor an Prof. Dr. F.X. Heiner, Hochwürden, 31. Dez. 1895]

As I have been told by a personally known brother monk of the author, P. Esser is momentarily in as a co-worker of an extremely important commission, appointed by the Holy Father in order to revise the Index [...] {{This "extremely important" commission renewed the index of prohibited books, i.e., those books which a devout Catholic was not allowed to read.}}

§ 385 Das Unendliche und die Theologie / Infinity and Theology (26)

Vom katholischen Standpuncte muß man froh sein, daß Sie den Herrn Prof. Riehl [...] los werden und man kann nur wünschen, daß kein Gesinnungsgenoße von ihm an seine Stelle tritt. Denn diese Sorte ist im Stande, viel Unheil anzurichten, wie Sie es ja an Riehl selbst durch lange Jahre hin erfahren haben. [...] Die Kieler Theologen mögen selbst sich überzeugen, was sie an ihm haben und mögen zusehen, wie sie mit ihm fertig werden. Außerdem können wir nicht wißen, ob nicht etwa die Göttliche Vorsehung gerade solche radikalen Leute den protestantischen Universitäten aus dem Grunde zuweisen läßt, damit der Zersetzungs- und Auflösungsprozeil des Protestantismus dadurch beschleunigt werde. Hätten wir wohl ein Intereße daran, dies zu verhindern? Mit Nichten! [...] Die Thatsache, daß ein Schüler und Freund des Herrn Prof. Riehl (D. Förster) wegen Majestätsbeleidigung verurtheilt worden ist, müßte der Großherzoglich Badischen Regierung auf privatem Wege (durch den Ihnen befreundeten Abgeordneten) als ein Grund vorgehalten werden, den von Prof. Riehl empfohlenen Candidaten mit dem grössten Misstrauen zu begegnen. Im Senat dürfte es besser sein, diesen Gesichtspunct nicht zu berühren. [Cantor an Prof. Dr. F.X. Heiner, Hochwürden, 31. Dez. 1895]

From the catholic point of view we have to be happy that you got rid of Prof. Riehl [...] and one can only wish that he will not be replaced by a like-minded person. Because this sort of men is able to cause much damage, as you have experienced with Riehl over many years. [...] The theologians of Kiel may convince themselves what they have got and may look how they can live with him. Further we cannot know whether divine Providence places just such radical in Protestant universities in order to accelerate the undermining and decay of Protestantism. Would we be interested to hinder that? Not at all! [...] The government of the grand duke of Baden should be informed in a private way (by your friend, the member of parliament) of the fact that a pupil and friend of Prof. Riehl (D. Foerster) has been sentenced because of lèse-majesté. This should be a reason to meet the candidate recommended by Prof. Riehl with greatest suspiciousness. In the senate it may be preferable not to touch this point.

§ 386 Das Unendliche und die Theologie / Infinity and Theology (27)

Sollten Sie Zeit haben, meine Arbeiten zu lesen; so werden Sie vielleicht finden, dass sehr wenig mathematische Vorkentniße zum Verständniß derselben erforderlich sind. [Cantor an Hemann, 28. Juli 1887]

Mit Bezug auf die Frage des actualen Unendlichen in creatis wiederhole ich zunächst, was ich Ihnen vor einem Jahre schrieb, daß eine gelehrte Vorbereitung in der Mathematik zum Verständniß meiner betreffenden Arbeiten ganz und gar nicht nöthig ist, so daß ein genaues Studium der letzteren dazu ausreicht. Jeder, und vor allen der geschulte Philosoph ist in der Lage, die Lehre vom Transfiniten zu prüfen und sich von ihrer Richtigkeit und Wahrheit zu überzeugen. [Cantor an Hemann, 2. Juni 1888]

Should you have time to read my papers, you might find that very little previous mathematical knowledge is required for the understanding. [Cantor to Hemann, 28 July 1887]

With respect to the question of the actual infinite in creatis I repeat first of all, what I wrote you one year ago. For the understanding of my relevant papers a scholarly preparation in mathematics is not at all necessary but a careful study is sufficient. Everyone and in particular the trained philosopher is able to scrutinize the teaching of the infinite and to convince himself from its correctness and truth. [Cantor to Hemann, 2 June 1888]

§ 387 Das Unendliche und die Theologie / Infinity and Theology (28)

Daß S. Thomas der auf Aristoteles zurückführenden Schulmeinung in Bezug auf die actual unendlichen Zahlen nur mit grossen Zweifeln und halben Herzens anhing, läßt sich mit Sicherheit feststellen [...] Denn die Thomassche Doctrin "mundum incepisse sola fide tenetur nec demonstrative probari posse" [Daß die Welt angefangen hat, wird nur im Glauben festgehalten, und es ist nicht möglich, dies durch einen Beweis zu begründen.] findet sich bekanntlich nicht bloß in jenem opusculo, sondern auch [...] noch an vielen anderen Stellen. Diese Doctrin wäre aber unmöglich, wenn der Aquinate den Satz "es gibt keine actual unendlichen Zahlen" für erwiesen gehalten hätte. Denn aus diesem Satz (wenn er wahr wäre) würde demonstrative mit größter Evidenz folgen, daß eine unendliche Zahl von Stunden vor diesem Augenblick nicht verflossen sein könnte; Es würde also das Dogma vom Weltanfang (vor endlicher Zeit) nicht als bloßer Glaubenssatz haben vertheidigt werden können. [Cantor an Hemann, 2. Juni 1888, entnommen einschließlich der Übersetzung der lateinischen Passage aus C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 380f]

It can be absolutely ascertained that St Thomas only with great doubts and half-heartedly adhered to the received opinion concerning the actually infinite numbers, going back to Aristotle. Thomas' doctrine "It can only be believed but it is not possible to be proven that the world has begun" is known to appear not only in that opusculo but also [...] in many other places. This doctrine however would be impossible if the Aquinatus had thought that the theorem "there are no actually infinite numbers" was proven. Because from this theorem, it would immediately follow with greatest evidence that an infinite number of hours could not have passed before the present moment. The dogma of a beginning of the world could not have been defended as a pure doctrine of belief.

§ 388 Das Unendliche und die Theologie / Infinity and Theology (29)

Es hat der paganistisch falsche Satz {{Es gibt keine actual unendlichen Zahlen}} auch ohne die Eigenschaft eines von der Kirche anerkannten Dogmas zu besitzen oder je besessen zu haben, vermöge seiner dogmenähnlichen Verbreitung unermeßlichen Schaden der christlichen Religion und Philosophie verursacht und man kann es daher dem heiligen Thomas von Aquino, meines Erachtens, nicht hoch genug zu Dank anrechnen, daß er diesen Satz als durchaus zweifelhaft auf‘s Deutlichste gekennzeichnet hat. [Cantor an Hemann, 21. Juni 1888, nach C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 384]

Zum Vergleich: Thomas von Aquin schreibt in der Summa Theologica I, Q. 7, A. 4: Nulla autem species numeri est infinita, quia quilibet numerus est multitudo mensurata per unum. Unde impossibile est esse multitudinem infinitam actu, sive per se, sive per accidens. Item, multitudo in rerum natura existens est creata, et omne creatum sub aliqua certa intentione creantis comprehenditur, non enim in vanum agens aliquod operatur. Unde necesse est quod sub certo numero omnia creata comprehendantur. Impossibile est ergo esse multitudinem infinitam in actu, etiam per accidens. Sed esse multitudinem infinitam in potentia, possibile est. http://www.corpusthomisticum.org/sth1003.html

Keine Sorte von Zahlen ist unendlich; denn jede Zahl ist eine durch die Eins zu messende Vielheit. Also kann es unmöglich, an sich oder zufällig, eine aktual unendliche Vielheit geben. Desgleichen ist jede Vielheit, die in der Natur der Dinge existiert, geschaffen; und jedes Geschaffene unterliegt einer bestimmten Absicht des Schaffenden; denn kein Wirkender wirkt ziellos. Also ist es notwendig, daß alles Geschaffene unter eine ganz bestimmte Zahl fällt. Daher kann es unmöglich eine aktual unendliche Vielheit geben auch nicht per Zufall. Aber eine potentiell unendliche Vielheit ist möglich.

Ja, wir sind dem Aquinaten für diese klaren Worte zu Dank verpflichtet!

The pagan, wrong proposition {{There is no actually infinite number}}, even without possessing the property of being a dogma acknowledged by the church or ever having been in that possession, has, because of its dogma-like popularity, done unmeasurable damage to Christian religion and philosophy, and one cannot, in my opinion, thank holy Thomas of Aquino too effusively that he has clearly marked this proposition as definitely doubtful. [Cantor to Hemann, 21 June 1888, quoted from C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 384]

For comparison: Thomas Aquinatus writes in his Summa Theologica I, Q. 7, A. 4: But no species of number is infinite; for every number is multitude measured by one. Hence it is impossible for there to be an actually infinite multitude, either absolute or accidental. Likewise multitude in nature is created; and everything created is comprehended under some clear intention of the Creator; for no agent acts aimlessly. Hence everything created must be comprehended in a certain number. Therefore it is impossible for an actually infinite multitude to exist, even accidentally. But a potentially infinite multitude is possible. http://www.sacred-texts.com/chr/aquinas/summa/sum010.htm

Yes we have to be grateful for those clear words!

§ 389 Das Unendliche und die Theologie / Infinity and Theology (30)

Es wird von Ihnen das Verhältniß der beiden Sätze I. "Die Welt hat sammt der Zeit vor einem endlichen Zeitabschnitt angefangen oder was dasselbe sagt: die bisher verflossene Zeitdauer der Welt ist (mit dem Maaß etwa einer Stunde gemeßen) eine endliche." welcher wahr und christlicher Glaubenssatz ist und: II. "Es giebt keine actual unendlichen Zahlen." welcher falsch und heidnisch ist und daher kein christlicher Glaubenssatz sein kann, ich sage, es wird von Ihnen das Verhältniß dieser beiden Sätze nicht richtig gedacht. [...] Aus der Wahrheit des Satzes I folgt aber mit Nichten, wie Sie in Ihrem Briefe anzunehmen scheinen, die Wahrheit des Satzes II. Denn der Satz I bezieht sich auf die concrete creatürliche Welt; Satz II aber auf das ideale Gebiet der Zahlen; im letzteren könnte das actual Unendliche vertreten sein, ohne deshalb in jener nothwendig vorkommen zu müßen. {{Das ist falsch. Jede Zahl besitzt wie jeder Gedanke eine physikalische Darstellung als Elektronenkonfiguration in einem Gehirn.}} [Cantor an Hemann, 21. Juni 1888, entnommen aus C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 383]

Your understanding of the relation of the two propositions I. "The world including the time has begun before a finite time interval or, what is the same, the duration of the world elapsed until now (e.g., measured by hours) is finite." which is true and a Christian dogma and II. "Actually infinite numbers do not exist." which is false and pagan and therefore cannot be a Christian dogma, I say you have not the correct idea about the relation of these two propositions. [...] The truth of proposition I does not at all imply, as you seem to assume in your letter, the truth of proposition II. Because proposition I concerns the concrete world of creation; proposition II concerns the ideal realm of numbers; the latter could include the actually infinite without its necessarily being included in the former. {{That is wrong. Every number has, like every thought, a physical representation as an electron configuration in a brain.}}

§ 390 Das Unendliche und die Theologie / Infinity and Theology (31)

Die Lehre vom Transfiniten ist weit davon entfernt, die Thomassche Doctrin in ihren Fundamentenzu erschüttern. Dagegen wird meine Lehre in gar nicht so ferner Zeit als eine geradezu vernichtende Waffe gegen allen Pantheismus, Positivismus und Materialismus sich erweisen! {{Cantor als Terminator.}} [Cantor an P. Joseph Hontheim S. J., 21. Dez. 1893]

Es besteht wohl volle Übereinstimmung darüber, dass Matheologie und Materialismus absolut unvereinbar sind. Matheologie erfordert den Glauben an eine platonistische immaterielle Existenz aller Mengen (freilich ohne die Menge aller Mengen und auch ohne die Menge aller Mengen ohne die Menge aller Mengen und auch ohne die Menge aller Mengen ohne die Menge aller Mengen ohne die Menge aller Mengen und auch ohne ...). Wie Engel im Himmel und Seelen im Irgendwo müssen Zahlen ohne jedes rationale Fundament existieren - die meisten ohnehin undefinierbar, unidentifizierbar, unerkennbar und natürlich unbenutzbar, also unbrauchbar. Der Materialismus dagegen lehnt die Existenz immaterieller, unphysikalischer Objekte ab. Es handelt sich um den allgegenwärtigen Widerstreit zwischen Glühen und Wissen, zwischen Religion und Wissenschaft.

The teaching of the transfinite is far from shaking the fundaments of the doctrin of Thomas. The time is not far, however, that my teaching will turn out to be a really exterminating weapon against all pantheism, and materialism. {{Cantor, the terminator!}}

I think we all can agree that matheology and materialism are absolutely incompatible. Matheology is impossible without the credo in a platonist immaterial existence of all numbers and all sets (without the set of all sets though and without the set of all sets without the set of all sets and without the set of all sets without the set of all sets without the set of all sets and without ...). Like angels in the heaven and souls in the somewhere numbers have to exist without any rational foundation - most of them even remaining undefinable, unidentifyable, unrekognizable, and unusable, i.e., useless. Materialism does not accept the existence of immaterial, unphysical objects. We see the eternal conflict between burning and knowing, between religion and science.

§ 391 Das Unendliche und die Theologie / Infinity and Theology (32)

Besonders kühn für seine Zeit, in dieser Sache, erscheint Rod. Arriaga S. J. [ SJ (1592 - 1667)]. (Ich bemerke hierbei, daß die Lehre vom creatürlichen actualen Unendlichen (was ich Transfinitum nenne) bei Rod. Arriaga keineswegs widerspruchsfrei begründet ist; dasselbe gilt von dem Minimen Em. Maignan [Emanual MAIGNAN OMin (1601 - 1676)]. Beide habe ich erst kennen gelernt lange nachdem ich meine Theorie innerlich fertig und in‘s Klare gebracht hatte. Es fehlt auch beiden die richtige Begriffsbildung der transfiniten Cardinalzahlen (Mächtigkeiten) und der transfiniten Ordnungstypen und Ordnungszahlen, also gerade dasjenige Instrument, mit dessen Hülfe die ganze Lehre einwandsfrei wird.); aber auch Suarez S. J. [Francisco SUAREZ SJ (1548 - 1617)] steht mir nicht so fern, wie es vielleicht den Anschein hat. [...] Bei meiner Hochschätzung und Verehrung Ihres religiösen Ordens könnte ich von keiner Seite mehr Ermuthigung ziehen, in meiner Arbeit fortzufahren, als von Ihnen und den Ihrigen! [Cantor an P. Joseph Hontheim S. J., 21. Dez. 1893, nach C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 396f] {{Die Wurzeln der Mengenlehre reichen in Zeiten zurück, in denen Giordano Bruno und wegen Ketzerei verurteilt wurden.}}

Especially bold in this matter, with regard to his time, appears Rod. Arriaga S. J. [Rodrigo de ARRIAGA SJ (1592 - 1667)]. (I mention here that the teaching of the creational actual infinite (what I call transfinitum) by Rod. Arriaga has not at all been founded free of contradiction; same is true for the Minime Em. Maignan [Emanual MAIGNAN OMin (1601 - 1676)]. Both I have only become acquainted with a long time after I had completed my theory internally and had cleared it. Both of them are lacking the correct notions of transfinite cardinal numbers (Maechtigkeiten) and the transfinite order types and ordinal numbers, just that tool which helps to make the whole theory faultless.); but also Suarez S. J. [Francisco SUAREZ SJ (1548 - 1617)] is not so disconnected from my position as it might appear. [...] With respect to my high esteem and admiration of your religious order I could not win more encouragement from any party to continue in my work than from you and yours. {{The roots of set theory reach back into times which saw Giordano Bruno and Galileo Galilei sentenced as heretics.}}

§ 392 Das Unendliche und die Theologie / Infinity and Theology (33)

Von Leibniz beispielsw. ist es sicher, daß er e. creatürl. Unendl. i. verschied. Beziehungen als wirklich existirend angenommen hat. [...] Dagegen hat Leibniz sowenig wie seine Vorgänger u. Nachfolger die act. unendl. d. h. transf. Zahlen u. Ordnungstypen erkannt; er bestreitet sogar ihre Möglichkeit. [Cantor an P. Ignatius Jeiler, OFM, 20. Mai 1888, nach C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 412]

It is certain that for instance Leibniz has assumed the creational infinite in different relations as really existing. [...] On the other hand Leibniz has as little as his predecessors and successors recognized the actually infinite transfinite numbers and order types; he even refutes their possibility. {{For further reading see § 20 and § 21.}}

§ 393 Das Unendliche und die Theologie / Infinity and Theology (34)

Während das hier Hervorgehobene (daß der Satz "totum majus est sua parte" in gewissem Sinne falsch ist) in Bezug auf die Substanzialformen allgemein anerkannt ist (beispielweise bleibt die Seele eines lebenden Organismus beim Wachsen oder Abnehmen des Körpers ihrem wesentlichen Sein nach stets dieselbe {{ein wichtiger Aspekt im Reiche des aktual Unendlichen}}), scheint man zu glauben, daß es für die accidentalen Formen nicht auch zutreffe. Dieses Vorurtheil ist eben aus der Wahrnehmung entstanden, daß, wie ich soeben hervorhob, bei endlichen Mengen, auf die man allein seine Betrachtungen beschränkt hatte, der Satz "tot. e. majus sua parte" in Bezug auf die diesen Mengen zukommenden Cardinalzahlformen stets richtig ist; ohne weitere Untersuchung, aber auch ohne jegliche Berechtigung, wurde seine Gültigkeit im bezeichneten Sinne auch auf unendliche Mengen übertragen, und man darf sich daher über die Widersprüche nicht wundern, die aus einer so grundfalschen Voraussetzung sich ergaben. [Cantor an P. Ignatius Jeiler, OFM, 20. Mai 1888, nach C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 418]

Classical logic was abstracted from the mathematics of finite sets and their subsets [....] Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. [Hermann Weyl: "Mathematics and logic: A brief survey serving as a preface to a review of The Philosophy of Bertrand Russell", American Mathematical Monthly 53 (1946) 2-13.]

Warum ist es falsch, das Prinzip "totum majus est sua parte" auf unendliche Mengen zu übertragen, nicht aber die Prinzipen der klassischen Logik, z.B. die Implikation, die u.a. auf diesem Prinzip basiert?

Whereas the emphasized (that the principle "totum majus est sua parte" is wrong in a certain sense) with respect to substantial forms is acknowledged in general (the soul of a living organism, for instance, in its essential being remains always the same during the growing or decreasing of the body {{an important aspect in the realm of the actually infinite}}) one seems to believe that this does not refer to accidencial forms. This prejudice originates from the observation that, as I just stressed, observation has been restricted to only finite sets which always obey the principle "totum majus est sua parte" with respect to the cardinal number forms belonging to them; without further investigation, but also without any justification, its validity has been carried over to infinite sets, and there is no reason to be surprized about the contradictions resulting from such an utterly wrong premise. [Cantor]

Classical logic was abstracted from the mathematics of finite sets and their subsets [....] Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. [Weyl]

Why is it wrong to carry over to infinite sets the principle "totum majus est sua parte" but not the principles of classical logic, for instance the implication that, among others, is based upon this principle?

§ 394 Das Unendliche und die Theologie / Infinity and Theology (35)

Sie sagen [...] daß Ihnen d. Begriff des Transfiniten Schwierigkeiten verursache, weil Sie den Satz nicht aufgeben könnten, daß, wo additio möglich, ein Potenzielles vorhanden sein muß. Es ist aber von mir nicht behauptet. worden, daß ein Transfinitum nur Act. sei, vielmehr ist das Transf. in dem Sinne Potenz, in welchem es vermehrbar ist; nur das Absolute ist actus purus oder vielmehr actus purissimus. [Cantor an P. Ignatius Jeiler, OFM, 20. Mai 1888, nach C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 417]

Die Menge aller Zeilen einer Cantor-Liste ist vermehrbar. Also ist auch die Menge der sie nummerierenden natürlichen Zahlen vermehrbar. Also ist eine nicht in der Liste vorkommende Diagonalzahl kein Indiz für eine größere Mächtigkeit der reellen Zahlen sondern lediglich dafür, dass die Menge der natürlichen Zahlen abermals und beliebig oft verdoppelt werden kann. Tatsächlich könnte sie bei ausreichender Geduld und dem Vorhandensein des aktual ¡0 Unendlichen auch aktual unendlich oft verdoppelt werden, womit sich |Ù| ¥ ¡0ÿ2 ergäbe.

You say [...] that you have problems with the notion of the transfinite because you cannot give up the theorem that the possibility of additio implies the presence of a potential. But it has not been asserted by me that a transfinitum be only act rather the transfinite in this sense is potency in which sense it is multiplyable; only the absolute is actus purus or rather actus purissimus.

The set of all entries of a Cantor-list can be multiplied. Therefore also the set of numbers enumerating the entries can be multiplied. Therefore the diagonal number resulting from the list does not indicate a greater cardinality of the real numbers but only that the set of natural numbers can be doubled again and again, as often as desired. In fact, given sufficient patience and given that there are ¡0 steps possible, the natural numbers could be multiplied infinitely ¡0 often to get as many as |Ù| ¥ ¡0ÿ2 .

§ 395 Das Unendliche und die Theologie / Infinity and Theology (36)

Um scholastisch zu reden: was weiterer Vermehrung fähig ist, ist in potentia zu diesem weiteren actus, also ein potentielles; fällt also unter diesen Begriff. Ihr transfinitum könnte also hiernach nur eine Unterabtheilung des gewöhnlich gelehrten potentiellen Infiniten sein. [Jeiler an Cantor, 22. Juni 1890]

Betrachte die folgende Liste: 1 1 2 11 3 111 ......

Die natürlichen Zahlen 1, 2, 3, ... sind natürlich keiner Vermehrung fähig. Die natürlichen Zeilen 1, 11, 111, ... sind natürlich einer Vermehrung fähig. Man muss das nur richtig verstehen. Alle Elemente aller abzählbaren Mengen können (durch natürliche Zahlen) nummeriert werden und können durch die sie nummerierenden natürlichen Zahlen bezeichnet werden. Die natürlichen Zahlen bilden demnach die umfangreichste Menge von allen Mengen, die ausschließlich definierbare Elemente enthalten. Und andere kommen in der Mathematik nicht vor.

To use scholastic terms: Something that can be multiplied is in potentia to this further actus, hence something potential; it belongs to that notion. Your transfintum could be some subsection of the usually taught potential infinite.

Consider the following list: 1 1 2 11 3 111 ......

The natural numbers 1, 2, 3, ... are not capable of natural multiplication. The natural rows 1, 11, 111, ... are naturally capable of multiplication. You have to understand that only. All elements of all countable sets can be enumerated by (natural numbers) and can be denoted by the natural numbers enumerating them. The natural numbers therefore are the most extensive set of all sets that contain exclusively definable elements. And others do not appear in mathematics.

§ 396 Das Unendliche und die Theologie / Infinity and Theology (37)

Die Resultate, zu denen ich gelangt bin, sind diese: Ein solches Transfinitum, sowohl wenn es in concreto, wie auch in abstracto gedacht wird, ist widerspruchsfrei, also möglich und von Gott erschaffbar, so gut wie ein Finitum. [...] Alle diese besonderen Modi des Transfiniten existiren von Ewigkeit her als Ideen in intellectu divino. [...] Wenn Sie diese Thatsache so ausdrücken, daß Sie sagen: "jedes Transfinite ist in potentia zu einem weiteren actus und in sofern ein potenzielles", so ist nichts dagegen einzuwenden. Denn actus purus ist nur Gott; dagegen jedes Creatürliche, in dem von Ihnen gebrauchten Sinne, "in potentia zu einem weiteren actus sich befindet." Dennoch kann das Transfinite nicht als eine Unterabtheilung dessen angesehen werden, was man gewöhnlich "potentielles Unendliches" nennt. Denn letzteres ist nicht (wie jedes individuelle Transfinite und allgemein wie jedes Ding, das einer "Idea divina" entspricht) in sich bestimmt, fest und unveränderlich, sondern ein in Veränderung Begriffenes Endliches, das also in jedem seiner actuellen Zustände eine endliche Größe hat; Wie beispielsweise die vom Weltanfang verflossene Zeitdauer, welche, wenn man sie auf irgend eine Zeiteinheit, z. B. ein Jahr, bezieht, in jedem Augenblicke endlich ist, aber immerzu über alle endlichen Grenzen hinaus wächst, ohne jemals wirklich imendlich groß zu werden. [Cantor an P. Ignatius Jeiler, Ord. S. Franc., 13. Okt. 1895, nach H. Meschkowski: "Georg Cantor: Leben, Werk und Wirkung" 2. Aufl. BI, Mannheim (1981) p. 271f]

Hier ist ein Beipiel für beide Formen des Unendlichen: Betrachte die Folge oder Liste 0.0 0.1 0.11 0.111 ... Als Diagonalzahl kann (dkk) = 0.111... erzeugt werden. Wäre das Unendliche nur potentiell (in Veränderung Begriffenes Endliches - zu jeder Zeile gibt es eine nachfolgende) dann wäre die Diagonalzahl selbst in jedem ihrer "actuellen Zustände" in der Liste. Um eine Diagonalzahl zu erhalten, die sich von jeder Zahl der Folge unterscheidet, muss die Liste aktual unendlich viele Zeilen besitzen (die Zeilenzahl muss größer als jede natürliche Zahl sein). Da aber die Zeilen über die Diagonale mit den Spalten gekoppelt sind, ergibt sich ein Widerspruch. Die ersten Indizes k liefern eine aktual unendliche Diagonalzahl, die zweiten Indizes k eine nur potentiell unendliche.

The results, which I have arrived at, are as follows: Such a transfinite is free of contradiction, therefore possible and creatable by God, as well as a finite. [...] All special modes of the transfinite exist forever as ideas in intellectu divino. [...] If you express this fact by saying: "every transfinite is in potentia to another actus and thus is a potential", so I have no objection. Because actus purus is only God; but every creational, in the mentioned sense, is in potentia to another actus. Nevertheless the transfinite cannot be considered a subsection of that which is usually called "potentially infinite". Because the latter is not (like every individual transfinite and in general everything due to an idea divina) determined in itself, fixed, and unchangeable, but a finite in the process of change, having in each of its actual states a finite size; like, for instance, the time elapsed after the biginning of the world, which, measuered in some time-unit, for instance a year, is finite in every moment, but always growing beyond all finite limits, without ever becoming really infinitely large. [Cantor to P. Ignatius Jeiler, Ord. S. Franc., 13 Oct. 1895, fromH. Meschkowski: "Georg Cantor: Leben, Werk und Wirkung" 2nd ed. BI, Mannheim (1981) p. 271f]

Here we have an example for both kinds of the infinite: Consider the list or sequence 0.0 0.1 0.11 0.111 ... The diagonal number can be made (dkk) = 0.111... If infinity was only potential (always growing - after every row there follows another row), then the diagonal number was in the list "in each of its actual states". In order to get a diagonal number that is not in the list, the list need to have actually infinitely many terms (the number of rows must be larger than every natural number). Since horizontal and vertical rows are coupled by the diagonal, we get a contradiction. The first indices k supply an actually infinite diagonal number, the second indices k an only potentially infinite.

§ 397 Das Unendliche und die Theologie / Infinity and Theology (38)

{{An mindestens vier verschiedenen Stellen seiner erhaltenen Korrespondenz (und das ist leider nur ein kleiner Bruchteil der gesamten) betont Cantor:}} Zur Auffassung des Grundgedankens der Lehre des Transfiniten bedarf es keiner gelehrten Vorbildung in der neueren Mathematik; dieselbe kann dazu sogar hinderlich sein, weil in der sogenannten Infinitesimalanalysis das potenziale Unendliche sich in den Vordergrund gedrängt und selbst bei den Heroen die Meinung gezeitigt hat, als beherrschten sie mit ihren "Differentialen“ und "Integralen“ die Höhen des Wissens und Könnens. Strenggenommen ist aber überall das potenz. Unendl. ohne ein zu Grunde liegendes A. U. (über das sich nur die meisten jener Herren keine Rechenschaft geben mögen oder können) undenkbar. Wenn Sie also etwa in diesen Kreisen auf "fachmännisch" competentes Urtheil zu der vorliegenden Frage zählen sollten, so könnten Sie sich vielleicht getäuscht sehen; das alleinige Forum ist hier die höchstgebietende Vernunft, welche kein Ansehen der privilegirten, gelehrten, akademischen Zünfte anerkennt; sie bleibt und herrscht, wir Menschen kommen und gehen. [Cantor an Prof. Aloys Schmid, 18.? April 1887 nach C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 504]

{{At least in four of his preserved letters (and that is merely a small fraction of his complete correspondence) Cantor emphasizes:}} To understand the teaching of the transfinite no scholarly education in newer mathematics is required; this could even be a nuisance because in the so-called infinitesimal analysis the potential infinite has pushed to the fore and lead to the opinion, even of the heroes, as if they with their "differentials" and "integrals" mastered the hights of knowledge and skill. Strictly speaking, the potential infinite is always unthinkable without the foundational A. I. (which only most of those gentlemen cannot account for). So, if you expect to get an "expertly" competent judgement from those circles you may find your expectations disappointed. The only forum here is the Empress Reason which does not acknowledge any reputation of priviledged, scholarly, academical guilds. She persists and rules - we humans come and go.

§ 398 Das Unendliche und die Theologie / Infinity and Theology (39)

Alle sogenannten Beweise (und es dürfte mir wohl keiner verborgen geblieben sein) gegen das geschöpfliche A. U. beweisen nichts, weil sie sich nicht auf die richtige Definition des Transfiniten beziehen. Die beiden für seine Zeit und auch heute noch kräftigsten und tiefsinnigsten Argumente des S. Thomas Aquinatus [...] werden hinfällig, sobald ein Princip der Individuation, Intention und Ordination actual unendlicher Zahlen und Mengen gefunden ist. {{Für undefinierbare Zahlen ist Individuation eine unerfüllbare Forderung.}} [Cantor an Aloys Schmid, 26. März 1887]

All so-called proofs (and I hardly may have missed anyone) against the creational Actual Infinite do not prove anything because they do not refer to the correct definition of the transfinite. The two, for their time and even today, strongest and profoundest arguments of St Thomas Aquinatus [...] are invalid as soon as a principle of individuation and ordination of actually infinite numbers and sets has been found. {{With respect to undefinable numbers individuation is an unattainable demand.}}

§ 399 Das Unendliche und die Theologie / Infinity and Theology (40)

Der R. P Ign. Carbonelle hat in seiner schönen Schrift "Les confins de la science et de la philosophie, 3e ed. t. I cap. 4" den Versuch gemacht, den Gerdilschen Beweis für den zeitlichen Weltanfang dadurch zu retten, dass er zwar sehr scharfsinnig und kenntnissreich den Satz vertheidigt: "Le nombre actuellement infini n'est pas absurd", aber demselben den harten, unbarmherzigen und dissonirenden Nachsatz giebt: "mais il est essentiellement indéterminé". Auf diesen Nachsatz würde er vielleicht verzichtet haben, wenn er schon damals meine Arbeiten gekannt hätte, die sich von Anfang an, seit bald zwanzig Jahren, fast ausschliesslich mit dem Beweis der Individuations-, Specifications- und Ordinationsmöglichkeit des actualen Unendlichen in natura creata beschäftigen. Mit jenem Nachsatz steht und fällt aber der vom R. P. Carbonelle unternommene mathematische Beweis für den zeitlichen Anfang der Schöpfung. Was endlich die dritte These Ihres geschätzten Schreibens betrifft, so bin ich ganz auf Ihrer Seite, wenn Sie mit Nic. v. Cusa sagen, daß "in Gott Alles Gott ist", wie auch, dass "die Erkenntnis Gottes objectiver Seits das Incommensurable nicht als commensurabel, das Irrationale nicht als rational zu erkennen vermag, weil die göttliche Allerkenntnis, wie die göttliche Allmacht nicht auf Unmögliches gehen kann." {{Dagegen verlangt die heutige Matheologie Unmögliches, nämlich die Individuation undefinierbarer Zahlen.}} [Cantor an Aloys Schmid, 26. März 1887]

The R. P. Ign. Carbonelle, in his beautiful essay "Les confins de la science et de la philosophie, 3e ed. t. I cap. 4", has tried to save Gerdil's proof for a temporal beginning of the world by very astutely and scholarly defending the proposition: "Le nombre actuellement infini n'est pas absurd", but adding the hard, merciless, and dissonant afterthought: "mais il est essentiellement indéterminé". Perhaps he would have refrained from that afterthought if he had known at his time my works already, which from its beginning, for meanwhile nearly twenty years, has been concerned with ways of individuation, specification and ordination of the actual infinite in natura creata. But the proof for a beginning of the world in finite time, undertaken by R. P. Carbonelle, stands or falls with just this afterthought . Finally, with respect to the third thesis of your esteemed letter I fully agree that you with Nic. of Cusa say that "in God all is God" as well as that "the cognition of God objectively cannot recognize the incommensurable as commensurable, cannot recognize the irrational as rational, because the divine omniscience as well as the divine omnipotence cannot give rise to the impossible". {{Alas, present matheology can give rise to the the impossible, namely die individuation of undefinable numbers.}}

§ 400

Eine Zahl ist definierbar durch ihre Ziffern oder durch ihr Bildungsgesetz. Die Menge aller definierbaren Zahlen ist abgeschlossen unter allen mathematischen Operationen. Die Menge aller definierbaren Zahlen ist eine Untermenge der abzählbaren Menge aller endlichen Definitionen. Die hierfür vorausgesetzte Sprache ist Deutsch. Jede endliche Definition in jeder beliebigen Sprache kann in die deutsche Sprache übersetzt werden. Schluss: Die Mathematik, sofern sie in Deutsch betrieben werden kann, enthält nur abzählbar viele Zahlen.

A number is definable by its digits or by its construction rule. The set of definable numbers is closed under all mathematical operations, i.e., every definable number leads always to other definable numbers. The set of all definable numbers is a subset of the countable set of all finite definitions. The basic language is English. Every definition in whatever language can be translated into English. Conclusion: Mathematics, as far as it can be expressed in English, contains only countably many numbers.

§ 401 Das Unendliche und die Theologie / Infinity and Theology (41)

Wenn hier gesagt wird, daß ein mathematischer Beweis für den zeitlichen Weltanfang nicht geführt werden könne, so liegt der Nachdruck auf dem Wort "mathematischer" und nur soweit stimmt meine Ansicht mit der des St. Thomas überein. Dagegen dürfte gerade auf Grund der wahren Lehre vom Transfiniten ein gemischter mathem. metaphysischer Beweis des Satzes wohl zu erbringen sein und insofern weiche ich allerdings von St. Thomas ab, der die Ansicht vertritt: S. th. q. 46, a. 2 concl. "Mundum non semper fuisse, sola fide tenetur, et demonstrative probari non potest." [Cantor an Aloys Schmid, 26. März 1887, nach H. Meschkowski, W. Nilson: "Georg Cantor Briefe", Springer, Berlin (1991) p. 285]

If it is said here that a mathematical proof of the beginning of the world in finite time cannot be given, the stress is on the word "mathematical" and only in that respect my opinion is in agreement with St. Thomas. On the other hand, based upon the true teaching of the transfinite, a mixed mathematical-metaphysical proof of the theorem might well be possible. In so far I differ from St. Thomas, who holds the opinion: "We know by belief only that the universe did not always exist, and that cannot be checked by proof on its genuineness."

§ 402 Das Unendliche und die Theologie / Infinity and Theology (42)

Ich halte es für sehr werthvoll, daß den schamlosen Angriffen Haeckels gegen das Christentum der angemasste Schein der Wissenschaftlichkeit nunmehr vor dem weitesten Kreisen entrissen wird. Die vornehme Scheu vor herzhafter Polemik (in anderen Kreisen so verbreitet!) musste gegenüber solchen Nichtswürdigkeiten weichen. [Cantor an Friedrich Loofs, 24. Feb. 1900]

I do highly appreciate that the pretended scientific appearance has been snatched away from Haeckel's shameless attacks against Christianity in front of the widest audience. The noble shyness toward hearty polemics (in other circles so usual!) had to give way with respect to such wretchedness.

§ 403 Das Unendliche und die Theologie / Infinity and Theology (43)

Als Philosoph thut man meines Erachtens überhaupt gut daran, sich in allen mathematisch- philosophischen Fragen mathematischen Autoritäten gegenüber möglichst skeptisch zu verhalten, eingedenk des wahren Pascalschen Ausspruchs: "Il est rare, que les géomètres soient fins, et que les fins soient géomètres." [Es ist selten, dass die Mathematiker Scharfsinnige sind und dass die Scharfsinnigen Mathematiker sind.] [Cantor an Aloys Schmid, 8. Mai 1887, nach C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 509]

As a philosopher you do well, in my opinion, to be very sceptical against mathematical authorities in all mathematical-philosophical questions, in memory of Pascal's true saying: "Il est rare, que les géomètres soient fins, et que les fins soient géomètres." {{It is rare that mathematicians are sharp-witted and that the sharp-witted are mathematicians.}}

§ 404 Das Unendliche und die Theologie / Infinity and Theology (44)

Die Thatsache der act. unendl. grossen Zahlen ist sowenig ein Grund für die Möglichkeit einer a parte ante unendlichen Dauer der Welt, dass vielmehr mit Hülfe der Theorie der transfiniten Zahlen die Nothwendigkeit eines von der Gegenwart in endlicher Ferne gelegenen Anfangs der Bewegung und Zeit bewiesen werden kann. Die ausführliche Begründung dieses Satzes verschiebe ich auf eine andre Gelegenheit, da ich Ihnen den frohen Ferienanfang nicht mit mathematisch-metaphysischen Erwägungen beschweren möchte. [Cantor an Aloys Schmid, 5. Aug. 1887, nach C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 517]

The fact of the act. infinitely large numbers is so little a reason for the possibility of an a parte ante infinite duration of the world that, on the contrary, by means of the theory of transfinite numbers the necessity of a beginning of and time in finite distance from the present can be proven. The detailled grounds of this theorem I will postpone to another opportunity because I would not like to weigh down the merry beginning of your holidays with mathematical and metaphysical considerations.

§ 405 Das Unendliche und die Theologie / Infinity and Theology (45)

Dagegen ist [...] in der ersten Hälfte des vorigen Jahrhunderts ein merkwürdiger Versuch von dem berühmten Franzosen Fontenelle gemacht worden (in dem Buche "Eléments de la Géometrie de l'infini", Paris 1727), actual unendliche Zahlen einzuführen; dieser Versuch ist jedoch gescheitert und hat dem Verfasser nicht ganz unverdienten Spott seitens der Mathematiker eingetragen, welche im 18ten Jahrh. und im ersten Viertel dieses Jahrh. gewirkt haben; die heutige Generation weiss nichts mehr davon. Fontenelle's Versuch musste scheitern, weil seine unendlichen Zahlen einen flagranten inneren Widerspruch mit sich auf die Welt brachten; es war leicht, diesen Widerspruch aufzudecken und ist dies von dem R. P. Gerdil bestens geschehen. Wenn aber d'Alembert, Lagrange und Cauchy geglaubt haben, dass damit die schlummernde Idee des Transfiniten tödtlich für alle Zeiten getroffen worden sei, so erscheint mir dieser Irrthum weit grösser, als der des Fontenelle und umso gravirender, als Fontenelle in der bescheidensten Weise sich als Laie in der Mathematik bekennt, während jene drei nicht nur Mathematiker von Fach, sondern wahrhaft grosse Mathematiker waren. [Cantor an Prof. Aloys Schmid, 26. März 1887, nach C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 500f]

In the first half of the last century a curious attempt has been made by the famous French Fontenelle (in the book "Eléments de la Géometrie de l'infini", Paris 1727), to introduce actually infinite numbers; this attempt however has failed and has brought him some mockery, not quite undeserved, from the mathematicians who were active in the 18th century and in the first quarter of this century; the present generation does not know about that. Fontenelle's attempt was doomed to failure because his infinite numbers brought with them a flagrant contradiction; it has been easy to show this contradiction, and that has been done best by R. P. Gerdil. But if d'Alembert, Lagrange and Cauchy have believed that the dormant idea of the transfinite has been stroken deadly by that for all times, then this error appears by far greater than that of Fontanelle and the more grave because Fontenelle in the most humble way confesses to be a layman in mathematics whereas those three have not only been professionals but really great mathematicians.

§ 406 Das Unendliche und die Theologie / Infinity and Theology (46)

Wenn man aber irgendetwas in der Wahrheit erkannt hat, so weiss man auch, dass man diese Wahrheit besitzt und man findet (auch wenn man, wie ich, sagen kann "non quaero ab hominibus gloriam" [von den Menschen verlange ich keinen Ruhm]) eine Art von Verpflichtung, soweit und solange die Kräfte dazu einem geschenkt werden, das Gewußte anderen mitzutheilen. Von diesem Gesichtspuncte aus wollen Ew. Hochwürden es freundlich entschuldigen, wenn ich im Folgenden ausführlicher das schon in meinen bisherigen Mittheilungen Gesagte vertreten und ergänzen werde. [Cantor an Prof. Aloys Schmid, 18.? April 1887, nach C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 504]

If one has recognized the truth of something, then one knows to be in possession of the truth and one feels (even if saying like me "non quaero ab hominibus gloriam" {{I do not want glory from mankind}} sort of duty, as far and as long as power reaches, to tell it to others. Under this aspect you, Reverend Father, will kindly forgive that I in the following will in greater detail support and amplify what I said in my recent messages.

§ 407 Das Unendliche und die Theologie / Infinity and Theology (47)

Nachdem ich erst kürzlich Ihr Werk "Institutiones philosophicae“ durchgesehen habe, bin ich zu der Ueberzeugung gelangt, dass ich in den allerwichtigsten metaphysischen Fragen von der Philosophie des heiligen Thomas Aquinatus, welche durch Ew. Hochwürden so meisterhaft und lichtvoll vertreten wird, nicht sehr abweiche und dass diejenigen Puncte, in welchen eine Differenz zu constatiren wäre, solche sind, welche eine Modification der Lehre des grossen Philosophen gestatten und vielleicht selbst wünschenswerth erscheinen lassen. [Cantor an Révérend Père Matth. Liberatore S. J., 7. Feb. 1886, nach C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 453]

After recently skimming through your paper "Institutiones philosophicae“ I got the impression that I do not much deviate in the most important metaphysical questions concerning the philosophy of Saint Thomas Aquinatus, which you, Referend Father, support so masterly and enlightening, and that those points, where a difference could be stated, are such in which modifying the teaching of the great philosopher might be allowed and perhaps even be desirable.

§ 408 Das Unendliche und die Theologie / Infinity and Theology (48)

Hinsichtlich der Frage, die du in deinen Arbeiten behandelst, bin ich der Ansicht, daß eine unendliche Vielheit als in Wirklichkeit existierende unmöglich, als im Denken existierende nicht nur möglich, sondern aktual im göttlichen Intellekt gegeben ist; denn sicher nimmt Gott alle möglichen Dinge unterschieden wahr, und es gibt unendlich viele mögliche Dinge. Ich glaube, daß dies die Lehrmeinung des heil. Thomas ist. lch stimme dir nicht zu im Hinblick auf die Theorie von den einfachen Seienden, die als konstitutive Prinzipien der Körper zugelassen werden. Die wesensmäßige Zusammensetzung der Körper kann meiner Meinung nach nur durch Materie und Form im Sinne des Aristoteles und der Scholastiker erklärt werden. Ich glaube, daß die Lehre des heiligen Thomas keine Modifikation in Bezug auf ihre fundamentalen Teile vertragen kann. Wenn irgendeiner ihrer Teile weggenonmnmen wird, stürzt das ganze Gebäude zusammen; so sehr ist jene Lehre in sich zusammenhängend. [Liberatore an Cantor, 24. Feb. 1886 {{Antwort auf § 407}}, Originaltext Latein, Übersetzung aus C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 454}]

With respect to the question that thou are treating in thy works, I mean that an infinite multitude as in reality existing is impossible, as in thinking existing is not merely possible but actually is being given in the divine intellect; surely God perceives all possible things as distinct, and there are infinitely many possible things. I believe that this is the doctrine of holy Thomas. I do not agree with you concerning the theory of the simple beings which are admissible as the constituent priciples of bodies. The essential constitution of bodies can, in my opinion, be explained only by means of matter and form in the sense of Aristotle and the scholastics. I believe that the doctrine of holy Thomas cannot bear any modification with respect to the fundamental parts. When any of its parts is removed, the whole edifice will collapse; so closely this teaching is interconnected.

§ 409 Das Unendliche und die Theologie / Infinity and Theology (49)

Zur Sache möchte ich noch hinzufügen, dass Sie in der bisherigen Mathematik, im Besonderen in der Differential- und Integralrechnung wenig oder gar keine Auskunft über das Transfinitum erhalten können, weil hier das potenziale Unendliche, ich sage nicht die alleinige, aber die an die Oberfläche (mit welcher sich die meisten Herren Mathematiker gern begnügen) hervortretende Rolle spielt. Selbst Leibniz, mit dem ich auch sonst in vielen Beziehungen nicht harmonire, hat sich [...] in Bezug auf das A. U. in die auffallendsten Widersprüche verwickelt, [Cantor an Aloys Schmid, 26. März 1887, nach C. Tapp: "Kardinalität und Kardinäle: Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit." Boethius Vol. 53, Franz Steiner Verlag (2005) p. 500]

On account of the matter I would like to add that in conventional mathematics, in particular in differential- and integral calculus, you can gain little or no information about the transfinite because here the potential infinite plays the important role, I don't say the only role but the role that is visible next to the surface (which most colleagues are readily satisfied with). Even Leibniz [...] from whom I deviate in many other respects too has fallen into most eclatant contradictions with respect to the actual infinite.

§ 410 Das Unendliche und die Theologie / Infinity and Theology (50)

Sanctissimo Domino Nostro Papae LEONI XIII Ad Epistolas Apostolicas Sanctitatis Tuae Henoticas cum spectarem imprimis ad illam, 14. Apr. anni 1895 datam, quam ad Populum Anglicum misisti, confessionem fidei Francisci Baconi «Seculi et gentis suae decoris, ornatoris et ornamenti literarum» Christianis Omnibus et praecipue Anglicanae Ecclesiae Sectatoribus in memoriam revocare opportunum existimavi. Permitte, Pontifex Maxime, ut septem exemplaria novae editionis hujus opusculi Sanctitati Tuae dedecem et ut tria volumina operum Francisci Baconi addam. Oro rogoque Te, Beatissime Pater, ut accepta habere velis haec decem munera, que offere audeo, ut signa sint meae reverentiae meique Amoris Tuae Sanctitatis et Ecclesiae S. Catholicae Romanae. Tuae Sanctitatis humillimus et addictissimus servus Georgius Cantor Mathematicus.

Unserem heiligsten Herrn Papst Leo XIII. In Betracht der wohlbekannten Briefe Ihrer Apostolischen Heiligkeit, besonders jenes, unter dem 14. April 1895 gegebenen, den Du an das englische Volk geschickt hast, habe ich es für nötig gehalten, das Glaubensbekenntnis des Francis Bacon, "seines Jahrhunderts und seines Volkes Zier, Schmücker und Schmuck der Gelehrsamkeit" allen Christen und insbesondere den Anhängern der Anglikanischen Kirche ins Gedächtnis zu rufen. Erlaube, Größter Brückenbauer, daß ich sieben Exemplare einer neuen Ausgabe jenes kleinen Werkes Dir widme, und daß ich drei Bände der Werke des Francis Bacon beifüge. Ich bete und bitte Dich, Seeligster Vater, daß Du annehmen wollest jene 10 kleinen Gaben, die ich anzubieten wage, die Zeichen sein sollen meiner Verehrung und meiner Liebe zu Deiner Heiligkeit und der Heiligen Katholischen Römischen Kirche. Deiner Heiligkeit demütigster und höchst zugetaner Diener Georg Cantor Mathematiker. Eine Antwort des Papstes ist nicht bekannt. [Cantor an Papst Leo XIII, 13. Feb. 1896, H. Meschkowski, W. Nilson: "Georg Cantor Briefe", Springer, Berlin (1991) p. 383]

Our Holiest Father, Pope LEO XIII With regard to the well-known Letters of your Apostolic Holiness. in particular that published on April 14, 1895 that you have sent to the English people, I have held it necessary to remind all Christians, in particular the adherents of the Anglican Church, of the creed of Francis Bacon "the fine specimen of his century and his nation, adorning literature and being its adornment". Permit, Greatest Pontifex, that I dedicate to you seven specimen of a new edition of that little work and that I include three volumes of the works of Francis Bacon. I further pray and ask you, Beatissime Pater, to accept those 10 litttle gifts, which I dare to offer to you and which shall be a token of my admiration and of my love to your Holiness and to the Holy Catholic Roman Church. Your Holiness most humble and devoted servant Georg Cantor mathematician. An answer of the Pope is not known.

§ 411 Ein neuer Überabzählbarkeitsbeweis / A fine proof of uncountability

Wir betrachten eine Abzählung aller rationalen Zahlen und bilden die Antidiagonalzahl d = 0,d1d2d3 ...dn... Die Menge der Dezimalstellen dn (Ziffer samt Index) steht in Bijektion mit der Menge der natürlichen Zahlen. Jede endliche Ziffernfolge 0,d1d2d3...dn besitzt unendlich viele Fortsetzungen, zum Beispiel 0,d1d2d3...dn000... oder 0,d1d2d3...dn111... Also existieren in der Abzählung bis zu jeder Dezimalstelle dn von d unendlich viele Zahlen mit derselben Dezimalstellenfolge. Da die Antidiagonalzahl aber von allen rationalen Zahlen verschieden ist, muss diese Verschiedenheit durch die Verschiedenheit von Dezimalstellen bewirkt werden. Und weil die Dezimalstellen mit endlichen Indizes dazu nicht taugen, muss mindestens eine weitere Dezimalstelle zu d gehören. Damit ist die Überabzählbarkeit der Folge von Dezimalstellen von d bewiesen. Der Beweis sollte leicht auf jede andere Folge übertragbar sein. Damit ist gezeigt, dass jede unendliche Menge überabzählbar ist.

The digits dn of the antidiagonal d are in bijection with Ù. The entries of a rationals-complete list do not all differ at a finite index n from the antidiagonal d. For every finite index n there are infinitely many duplicates of 0.d1d2d3...dn. So if d differs by its digits from all rationals of the rationals-complete list, then the digits of d must be uncountable. There must be at least one digit more than those which are in bijection with Ù. A fine proof of uncountability of sequences.

§ 412 What is the meaning of "to exist" in mathematics? There are three alternatives with respect to numbers: 1) A number exists if it can be individualized in mathematical discourse such that its numerical value can be calculated by everybody without any error. 2) A number exists if it can be individualized in mathematical discourse such that its numerical value can be calculated by everybody with error less than any given epsilon. 3) A number exists if it cannot be individualized but if there is some dubious proof saying that some numbers of some sort should exist. Are there any other definitions possible?

§ 413 Irrational numbers have no decimal (or binary or whatever integer-positive-base) expansion. It is impossible for an infinite list of decimals to appear in mathematical discourse, dialogue, or monologue other than as the finite rule how to calculate every decimal at a finite place but never all decimals, since beyond every finite index there are infinitely many further indices. Every decimal that appears in mathematical discourse, dialogue, or monologue belongs to a rational number. By the way that is also the reason why Cantor's uncountability proof must fail. A matheologian (for the definition see § 001) answered: "Both your assertions above are incorrect." Wouldn't that claim oblige him to support his opinion by listing all decimals of a nonterminating decimal representation of a real number of his choice?

§ 414 The adaptation of strong axioms of infinity is thus a theological venture, involving basic questions of belief concerning what is true about the universe. [A. Kanamori, M. Magidor: "The evolution of large cardinal axioms in set theory" in: Higher Set Theory, Lecture notes in mathematics 669, G.H. Müller und D.S. Scott (eds.), Springer, Berlin (1978) p. 104]

§ 415 I came to the conclusion some years ago that CH is an inherently vague problem [...]. This was based partly on the results from the metatheory of set theory showing that CH is independent of all remotely plausible axioms extending ZFC, including all large cardinal axioms that have been proposed so far. In fact it is consistent with all such axioms (if they are consistent at all) that the cardinal number of the continuum can be “anything it ought to be”, i.e. anything which is not excluded by König’s theorem. The other basis for my view is philosophical: I believe there is no independent platonic reality that gives determinate meaning to the language of set theory in general, and to the supposed totality of arbitrary subsets of the natural numbers in particular, and hence not to its cardinal number. Incidentally, the mathematical community seems implicitly to have come to the same conclusion: it is not among the seven Millennium Prize Problems established in the year 2000 by the Clay Mathematics Institute, for which the awards are $1,000,000 each; and this despite the fact that it was the lead challenge in the famous list of unsolved mathematical problems proposed by Hilbert in the year 1900, and one of the few that still remains open. [Solomon Feferman: "Philosophy of mathematics: 5 questions" p. 12] http://www.academia.edu/160395/Philosophy_of_mathematics_5_questions

By continuum hypothesis, CH, we understand the assumption that the set of real numbers ¡0 (falsely identified by Cantor with the geometric continuum) has cardinal number ¡1 = 2 . It is absolutely meaningless, first because the continuum is inherently different from the set of real numbers, second because all alephs, countability and uncountability, are ill-defined super tasks, and third because there is not the least application of CH in science or mathematics.

§ 416 Alien mathematics: is π universal?

Aliens evolving elsewhere would probably not be humanoid. There is one area, though, where aliens are generally expected to be much like us: mathematics. It is often suggested that a good way to contact extraterrestrials is to send signals with the prime numbers, or digits of π. Each Voyager spacecraft carried a golden phonograph record of sounds and images of Earth, including a description of our number system. [...] It seems silly to imagine that intelligent creatures would think that 2 + 2 is different from 4. But I’m not so sure that they would necessarily understand 2, 4, or +. Let alone π. History shows that our mathematics depends not just on logical universals, but on what sort of creatures we are, where we live, and what we think is important. [Ian Stewart: "Alien mathematics: is Pi universal?" (2010)] http://www.telegraph.co.uk/travel/7954877/Alien-mathematics-is-Pi-universal.html

Aliens would know that two aliens and two aliens give four aliens. Aliens would probably know that the circumference of the circle isn't a rational multiple of its diameter so that π cannot be expanded by digits. But I wish I could know what aliens think about transfinite set theory - after they were informed. I am quite sure that this kind of "mathematics" is unique in the universe.

§ 417 Eine Konsequenz des aktual Unendlichen / An implication of actual infinity

Jede Menge Sn der Folge S1 = {1} S2 = {1} » {2} S3 = {1} » {2} » {3} ... ergibt sich als Vereinigung von {n} und {1, 2, 3, ..., n-1}. Unendlich viele Vereinigungen führen zu unendlich vielen Mengen, doch keine Menge enthält unendlich viele (alle) natürlichen Zahlen. Vielmehr fehlen in jeder der Mengen unendlich viele. Nochmals: Die Folge enthält unendlich viele Vereinigungen. Mit jeder wächst die Menge der darin enthaltenen natürlichen Zahlen. Doch die Menge Ù aller natürlichen Zahlen ist als Grenzwert einer streng monoton wachsenden Folge nicht in der Folge enthalten. Vereinigen wir aber alle Folgenglieder Sn (also alle Fehlversuche, Ù zu erzeugen) ohne irgendetwas hinzuzufügen, dann erhalten wir die Menge Ù aller natürlichen Zahlen. Das ist eine Konsequenz des aktual Unendlichen. Ist sie akzeptabel?

Every set Sn of the sequence S1 = {1} S2 = {1} » {2} S3 = {1} » {2} » {3} ... emerges from the union of {n} and {1, 2, 3, ..., n-1}. There are infinitely many unions causing infinitely many sets but no set contains infinitely many (all) natural numbers. In every set infinitely many of all natural numbers are missing. I repeat, there are infinitely many unions, each one adds another natural number n, but the set Ù, as the limit of a strictly monotonic increasing sequence of sets, is not a term of this sequence. But if we union all the terms Sn of the sequence (i.e., all the successless attempts to establish Ù) without adding anything further, then we get the complete set Ù of all natural numbers. An implication of actual infinity. Are you happy to accept it?

§ 418 Would you trust in such a theory?

Consider a geometry that contains the axiom: "For every triple of points there exists a straight line containing them." When you ask for the straight line that contains the points (0,1), (0,2), (1,0) the masters of the theory reply that some straight lines cannot be constructed but that they certainly "exist". If you ask what in this case existence would mean, you are called a crank. Would you trust in such a theory and its masters? You do already. The axiom of choice says that every set can be well-ordered, i.e., all its elements can be indexed such that every non-empty subset has an element with smallest index. There are only countably many indices, but what about uncountable sets? Who cares! But even if you are not outwitted by this case, you certainly trust in set theory, don't you? The common interpretation of the notion "set" is that all its elements "exist". The axiom of infinity then says that every element of an inductive set is preceded by finitely many elements but followed by ¡0 elements. Nobody has ever succeeded to show one of the trailing ¡0 elements. All that could be done is to show elements belonging to first the 0% set. In fact the situation is not very different from the geometry with the straight lines through every triple of points. It is rather the same. If you are in despair now then you show that you are able to follow mathematical arguments. And I can comfort you: Set theory does not require that sets have to be completed, neither does the axiom of infinity. This axiom has the same wording in potential infinity (from where it originally has been taken) and does not require ¡0 but only infinitely many successors to every element of an inductive set. That is a big difference. And the axiom of choice is a very natural one and is quite right because there are no uncountable sets.

§ 419 Warum versagt das Cantorsche Diagonalisierungsverfahren?

Jede abbrechende Dezimalzahl gehört zur potentiell unendlichen Menge der abbrechenden Dezimalzahlen. Sie sind durch Auflistung aller ihrer Ziffern identifizierbar (weil nach der letzen von Null verschiedenen nur noch Nullen folgen, die den Zahlenwert unverändert lassen). Jede unendliche Dezimalzahl gehört zur aktual unendlichen Menge aller nicht abbrechenden Dezimalzahlen. Sie sind im mathematischen Diskurs nicht durch Auflistung ihrer Ziffern identifizierbar, sondern erfordern ein endliches Bildungsgesetz. Eine Cantor-Liste, die alle durch Ziffern identifizierbaren Zahlen enthält (was nach Cantor möglich ist), kann keine durch Ziffern identifizierbare Antidiagonalzahl enthalten, die sich von allen Zahlen der Liste unterscheidet. Ebenso kann der Binäre Baum keinen durch Auflistung seiner Knoten identifizierbaren Pfad enthalten, der sich von allen Pfaden des Binären Baums unterscheidet. In beiden Fällen erfordert die erfolgreiche Diagonalisierung die Bildung eines durch Auflistung von Ziffern bzw. Knoten nicht identifizierbaren Grenzwertes. Zum Ende des 19. Jahrhunderts, als Cantor sein Diagonalverfahren veröffentlichte, war noch nicht bekannt, dass nur abzählbar viele endliche Definitionen für Bildungsgesetze existieren können. (Tatsächlich existieren zu jedem Zeitpunkt sogar nur endlich viele.) Dies wurde erst zu Beginn des 20. Jahrhunderts einigen wenigen Mathematikern bewusst und erst zu Beginn des 21. Jahrhunderts ins allgemeine Bewusstsein gerückt. Damit ist die Existenz von überabzählbaren Zahlenmengen (für andere Mengen gelten ähnliche Überlegungen) im mathematischen Diskurs ausgeschlossen. Die Bezeichnungen für diese im mathematischen Diskurs nichtexistenten aber von vielen Mitgliedern der intellektuellen Arrièregarde noch immer für wahr genommenen Mengen existieren als endliche Definitionen weiterhin, haben aber keinen Bezug zur Mathematik.

Fußnote: Die Verfechter des Cantorschen Diagonalverfahrens behaupten, es sei ausreichend, dass für jede Listenzahl eine Ziffer existiert, die sich von der entsprechenden Ziffer der Antidiagonalzahl unterscheidet. Hier wird das Unendliche als potentiell behandelt. Das genügt aber nicht (es genügt nur scheinbar, wenn die Bedeutung des Unendlichen geflissentlich vertauscht und es nun als aktual behandelt wird - oder wenn man die Unterscheidung gar nicht kennt), wenn man die Antidiagonalzahl mit Hilfe ihrer Ziffern von allen Listenzahlen unterscheiden möchte, denn bis zu jeder (und also auch der gerade betrachteten) Ziffer der Antidiagonalzahl gibt es unendlich viele übereinstimmende Zahlen in einer Liste aller abbrechenden Dezimalzahlen. Um die Antidiagonalzahl von allen Listenzahlen zu unterscheiden (wie bereits bemerkt wird nur damit bewiesen, dass sie nicht in der aktual unendlichen Liste enthalten ist), genügen Ziffern allein also nicht. Dazu bedarf es einer endlichen Definition, aus der alle Ziffern der Antidiagonalzahl hervorgehen.

§ 419' Why does Cantor's diagonal argument fail?

Terminating decimal numbers can be identified by listing all their digits (because the last one different from zero is followed by zeros only which do not change the numerical value). Infinite decimal numbers cannot be identified in mathematical discourse by listing all their digits digit by digit. Each one requires a finite construction rule. A Cantor list containing all terminating decimal numbers cannot yield an antidiagonal that differs from all numbers of the list and can be identified by its digits. (Similary, the Binary Tree cannot be diagonalized such that the antidiagonal path differs from all paths of the Binary Tree and can be identified by its nodes.) In both cases the diagonal argument requires an infinite string, a limit that cannot be identified by listing all digits or nodes. A limit always requires a finite construction rule. At the end of the 19th century, when Cantor published his diagonal argument, it had not been known that only countably many finite words for expressing rules can exist*. It became known to few mathematicians only at the beginning of the 20th century and has become popular knowledge only at the beginning of the 21st century. This fact excludes the existence of uncountable sets of numbers in mathematical discourse, dialogue, or monologue.

The names for such sets, which nevertheless are yet asserted to exist somewhere (in a never defined place) by the intellectual arrièregarde, are persisting but without any relation to mathematics.

*) In order to eliminate Koenig's paradox of a first undefinable real number (which by these very words has been defined), it is usually argued that no list of all finite rules exists that could be followed by this number. However, that does not contradict the fact that the number of definable numbers is countable as is every subset of a countable set.

§ 420 Why mathematics contradicts set theory

In set theory there exists the list of all finite subsets of Ù. They can be noted in binary where the infinite sequence 000... is the empty set, 11000... is the set {1, 2}, and 111... is Ù (but the latter, as an infinite set, is not in the list). In set theory this list can be diagonalized. In mathematics the antidiagonal cannot be defined by its digits because all finite digit sequences are in the list and an infinite digit sequence cannot be communicated in mathematical discourse, dialogue, or monologue. Therefore the antidiagonal must be addressed by a finite definition like "antidiagonal of this particular (insert definition) list". The list of all infinite subsets of Ù, defined by finite definitions, e.g., in Latin letters (like "set of all even numbers") and noted as infinite binary sequences (like 010101...) is a sublist of the list of all finite definitions. But this sublist is subcountable. It has an antidiagonal, that is also defined by the finite definition "antidiagonal of this particular (insert definition) list". However, in set theory there is nothing subcountable. Every subset of any set exists and can be addressed. According to set theory there is no undefined subset of Ù, hence no subcountability as above. This means that in set theory, based upon mathematics, the axioms of extensionality and of power set are in contradiction with the list of all finite definitions of infinite subsets of Ù. In mathematics based upon the actual infinity of set theory there is subcountability.

In potential infinity, this contradiction cannot occur. "Antidiagonal of this particular (insert definition number n) list" can be inserted infinitely often since no list is ever completed.

§ 421 Für jede natürliche Zahl n gilt fraglos |{1, 2, 3, ..., n}| = n und falls ein Grenzwert X existiert, so gilt auch

lim |{1,2,3,...,nX }| = nX→ sowie lim nX= |{1,2,3,..., }| nX→ also |{1, 2, 3, ..., X}| = X

Die naheliegenden Konsequenzen

|{1, 2, 3, ..., ¡0}| = ¡0

|{1, 2, 3, ..., ω}| = ω

|{1, 2, 3, ..., ¶}| = ¶ gelten aber nicht. Jeder zünftige und vermutlich schon jeder zukünftige Adept der Mengenlehre hat gelernt, dass allein

|{1, 2, 3, ...}| = ¡0 zulässig und richtig ist.

§ 422 What is a definition of a number?

The definition of a number must allow transmitter and receiver in mathematical discourse, dialogue, and monologue to identify this one (1) number uniquely.

If the question was: "How can we define a number?, then the answer could only be: "A number can be identified by a finite string of symbols taken from a countable alphabet". There are many ways to do so. Every definable number has infinitely many finite definitions. Most are known for the number zero or 0 or 0.000..., because in addition to the three finite definitions just given there are lots of sequences with improper limit oo, each of which has a sequence of reciprocals with limit 0.

The set of finite strings, however, is countable. In order to get a set of uncountably many numbers, infinite strings of symbols are required (because uncountable countable lists are impossible, but unlistable alphabets cannot be learned or applied, i.e., uncountable alphabets are not alphabets). Such an infinite string must be capable of uniquely defining a number. It is not enough to distinguish the number from all its finite approximations like 1/9 = 0.111... can be distinguished from all its finite approximation 0.1, 0.11 and so on. That would require that 1/9 is already "given", of course by a finite definition. Infinite definitions cannot "be given". A distinction by digits is impossible because all digits belong to the set of finite approximations. It is only the property of being "non-terminating" that distinguishes 1/9 uniquely from all its approximations, but this property cannot be obtained from checking any digits but only from the finite definition. (An always negative result with always infinitely many further digits remaining to check can only be accepted as a final exclusion in an experimental science like physics. Mathematics requires final proofs!)

Although it is clear from the above argument, that a number cannot be defined by an infinite sequence of digits, it can be proved in addition that the set of all infinite sequences of digits is countable. For this proof consider the set of all infinite sequences of symbols, or, without loss of generality, the Binary Tree which contains as paths the allegedly uncountable set of real numbers in the unit interval. All infinite bit sequences that in a unique way define a real number of this interval (some numbers having even two such sequences) are paths in the Binary Tree. All possible paths that are defined by nodes only will be covered when each node is covered by at least one infinite path containing it. Since the number of nodes is countable, this is accomplished by countably many paths. More paths cannot be defined by nodes. But even covering every node by countably many paths would not result in using more than countably many paths.

This excludes the acceptance of an uncountable set of numbers in any mathematical theory that is free of contradictions.

§ 423 The purposes of the Binary Tree proof is to show first that there are not more than countably many infinite paths and second that there is no actually infinite path.

The universe of ZFC set theory contains uncountably many real numbers each of which has a unique place in both the natural order by size (trichotomy) and all well-orderings, at least one of which is existing. This implies that every real number has a unique definition such that using this definition will call one and only one real number like a magic spell. Since there are at most countably many finite definitions, every real number is said to have an infinite definition. Such a definition can be tfansformed into bits. A very convenient method to do so is to identify each definition of a real number of the unit interval with an infinite path in the complete infinite Binary Tree

0. / \ 0 1 / \ / \ 0 1 0 1 ...

This Binary Tree contains a countable set of nodes n and infinite path p which are infinite sets of nodes, as we will assume for the outset.

(1) The complete structure of the Binary Tree can be constructed by a countable set of infinite path p(n). Proof: Every path p(n) is constructed such that it begins at the root node "0." and contains the node n and then continues in a way that can be arbitrarily chosen and does not matter for the proof, since every following node m will be constructed for another time by the path p(m) containing it. Note that every node can be connected with the root node in one and only one way. This way, called the common finite initial segment f(n) of all those paths that contain node n, is uniquely defined by node n. If all nodes of a path p (like 0.111...) have been constructed by paths, then the path p itself may have been used or may not have been used. We must say either that p has been constructed since p as a set of nodes cannot be different from all its nodes. Or we must say that it is impossible to discern the path p by the set of its nodes but only by a finite definition (like "0.111..."). However finite definitions are not under investigation here. Therefore all possible infinite paths that can be distinguished by nodes belong to a countable set of paths.

(2) In fact it is impossible to distinguish a path p (like 0.111...) by its nodes because all its nodes belong to other paths (like 0.1000..., 0.110000..., 0.111000...) too. The sequence of these other paths (here those with a finite number of nodes 1) is infinite and contributes all infinitely many nodes of p. The limit of this strictly increasing sequence does not belong to the sequence. But it would, if it was a path in the Binary Tree that could be identified by its nodes. In addition it is clear that in mathematical discourse never an infinite set can be identified by listing all its elements.

Conclusion: All paths in the Binary Tree are potentially infinite sequences of finite initial segments of infinite paths. Down to every level j there are only countably many such sequences consisting of finite initial segments of less than j + 2 nodes. Since one sequence cannot have many limits, there cannot be more than countably many limits. This result is in agreement with the number of not more than countably many definable real numbers.

§ 424 Actual Infinity: We never get it - but we get it!

In actual infinity the set Ù of all natural numbers exists as a union such that no natural number is missing. Considering the union stepwise

{1} {2, 1} {3, 2, 1} ... {..., 3, 2, 1} = Ù we see that Ù does not appear in any enumerated line, i.e., it appears never, but it appears according to the motto: It doesn't matter that we never get it - if only we get it. So the sequence somehow has to reach, create, or complete its limit.

This case can be translated into analysis. If actual infinity applies here, then the sequence

0.000 0.1000... 0.11000... 0.111000...... 0.111... reaches, creates, or completes its limit with an actual infinity of digits too. However, this implies that Cantor's diagonal argument fails in case the antidiagonal d of the sequence is chosen to be d = 0.111... Of course d is not completed in any enumerated line but only in the infinite - alas there it is already welcomed by itself.

Usually set theorists deny that d belongs to the infinite list. Therefore the projection of d on the horizontal axis is never completed (that would require a completed line). But its projection on the vertical axis is completed. And from that part it is concluded in reverse that d is completed. Only by this incoherent argument it is possible for d to differ from every line.

Not necessary to mention that in analysis this limit is not created by digits. We have to use finite definitions for what we never get by digits. The above list does never reach, create, or complete a string of digits without a tail of infinitely many zeros. And in analysis "never" means never.

§ 425 Why Russell's Paradox is irrelevant in mathematics

We are, like Poincaré and Weyl, puzzled by how mathematicians can accept and publish such results {{like the Hausdorff Sphere Paradox}; why do they not see in this a blatant contradiction which invalidates the reasoning they are using? [...] Presumably, the Hausdorff sphere paradox and the Russell Barber paradox have similar explanations; one is trying to define weird sets with self-contradictory properties, so of course, from that mess it will be possible to deduce any absurd proposition we please. [E. T. Jaynes: "Probability Theory: The Logic of Science", (Fragmentary Edition of March 1996)] http://www-biba.inrialpes.fr/Jaynes/cappb8.pdf

There are other paradoxes like Socrates' I know that I know nothing or Cantor's diagonal argument which proves by producing a finitely defined real number that the countable set of finitely defined real numbers is uncountable. It belongs to the set of all logical tricks but is not in any way related to determine sizes of sets. All this stuff is simply an amusement based on self- reference. Some of these tricks even comprehend eternal truths: Nobody has ever checked the ¡0 digits of Cantor's diagonal that follow behind the last checked digit. Or: A sausage is better than nothing and nothing is better than the eternal rapture. Others are so false that even their negation is false: "This sentence contains seven words" is wrong. "This sentence does not contain seven words" is wrong too. But such musings should not be confused with mathematics.

§ 426 - Our dissatisfaction with Zermelo's axiomatic in the context of the reality of the Continuum is rooted in the fundamental sequentiality of its constructions, the sequentiality which implies the sequential causality of all what could be said about transfinites. The advanced principles of transfinite set theory are designed to overcome this sequentiality obstruction, but they cannot eliminate it. - Ultimately, all modern transfinite set theory represents only a well designed fantasy founded on Zermelo's axiomatic, the fantasy which pushes to their limits the rich constructionist faculties of this system. All adaptations of these fantasies to even very modest aspects of the Continuum realities remain absolutely unsatisfactory. - This is because, as we claim, the origins of the Continuum are outside all set-theoretical "explanations". [E. Belaga "From Traditional Set Theory - that of Cantor, Hilbert, Goedel, Cohen - to Its Necessary Quantum Extension", IHES/M/11/18 (2011) p. 24] http://preprints.ihes.fr/2011/M/M-11-18.pdf

§ 427 Is successful diagonalization of a potentially infinite list possible?

William Hughes has claimed that a potentially infinite list of finite definitions of potentially infinite binary sequences can be subject to diagonalization:

Let L' be a list of finite definitions of potentially infinite 0, 1 sequences. Let L be the list of potentially infinite 0, 1 sequences. Define a function of two integers f by f(n,m) = the mth digit defined by the nth element of L'. Let dL(n) = f(n,n) +1 (mod 2). Then dL is a finite definition. The potentially infinite sequence defined by the definition does not end. Do not confuse definitions with the potentially infinite 0, 1 sequences they define. Use induction to show that for every n in Ù dL is not the nth element of L'. Use an indirect proof to show there is no n in Ù such that dL is the nth element of L'. However, by definition, dL is in L'. Contradiction.

No. Why does this not contradict the mathematics of potential infinity? The list L' of every finite definition of numbers can (and must) contain in some line k: "Define a function of two integers f by f(n,m) = the mth digit defined by the nth element of L'. Let dL(n) = f(n,n) +1 (mod 2). Then dL is a finite definition." Further every initial segment of the first j bits of the sequence referenced in line k will be defined in some line of L'. Insofar the list L' is as complete as a potentially infinite list can be. The anti-diagonal sequence of L, however, as a potentially infinite sequence of digits does not define anything since there is always a necessary piece of information missing (in fact even infinitely many). This shows that the potentially infinite set of finite definitions of sequences can not be successfully diagonalized.

§ 428 [...] we cannot have an infinite amount of rooms, this is the same as having an infinite amount of numbers - we cannot, we may keep counting all we like yet never get any closer to an infinite amount. [Amorphos: "Disproving infinity paradoxes; Hilbert's Hotel?" 17 June 2007] http://www.twcenter.net/forums/showthread.php?105283-disproving-infinity-paradoxes- Hilbert%92s-Hotel

It isn't just impossible "for us men" to run through the natural numbers one by one; it's impossible, it means nothing. [...] you can't talk about all numbers, because there's no such thing as all numbers. [L. Wittgenstein: "Philosophical Remarks" (1975)] http://www.press.uchicago.edu/ucp/books/book/chicago/P/bo3615160.html

§ 429 A point of view which the author feels may eventually come to be accepted is that CH is obviously false. The main reason one accepts the axiom of infinity is probably that we feel it absurd to think that the process of adding only one set at a time can exhaust the entire universe. Similarly with the higher axioms of infinity. Now ¡1 is the cardinality of the set of countable ordinals, and this is merely a special and the simplest way of generating a higher cardinal. The set C [the continuum] is, in contrast, generated by a totally new and more powerful principle, namely the power set axiom. It is unreasonable to expect that any description of a larger cardinal which attempts to build up that cardinal from ideas deriving from the replacement axiom can ever reach C. Thus C is greater than ¡n, ¡ω, ¡a, where a = ¡ω, etc. This point of view regards C as an incredibly rich set given to us by one bold new axiom, which can never be approached by any piecemeal process of construction. Perhaps later generations will see the problem more clearly and express themselves more eloquently. [P. Cohen: "Set Theory and the continuum hypothesis" Dover Publications (2008) p. 151] http://en.wikipedia.org/wiki/Paul_Cohen_(mathematician)

The continuum is not a set of co-ordinates. This has already become obvious by Cantor's proof of the same cardinality of the points in a femtometer interval and the whole universe. If someone does not see the inequality, it is deplorable but does not change the facts.

§ 430

Jede wohldefinierte Antidiagonale einer Cantor-Liste gehört zur abzählbaren Menge der wohldefinierten reellen Zahlen und, als endlicher Ausdruck, zur abzählbaren Menge der endlichen Ausdrücke. Eine undefinierte Folge von Dezimalziffern einer Antidiagonale stellt keine reelle Zahl dar, sondern höchstens ein Intervall, das sich erst im Unendlichen, also niemals, zu einem Punkt zusammenzieht. Merke: Jede versuchte mathematische Definition ohne Endsignal ist ungültig.

Every well-defined antidiagonal of a Cantor list belongs to the countable set of well-defined real numbers and, as a finite expression, to the countable set of finite expressions. An undefined sequence of digits of an antidiagonal does not represent a real number but at most an interval that only in the infinite, i.e. never, contracts to a point. Note: An attempted mathematical definition without end signal is invalid!

§ 431 Dear Sir, before continuing our discussion, I beg you to give a (formal or reasonable) explanation of the assertion that set theory allows or requires the use of all natural numbers in mathematical discourse or in any mathematical application. In my opinion, the axiom of infinity claims just the contrary, namely: Every natural number belongs to a finite initial segment of the sequence of natural numbers, and beyond every natural number there are infinitely many following. It is impossible to derive the use of all natural numbers from the definition of the notion "set" because such a definition does not exist. Without any axiomatic or other foundation it is assumed that all ¡0 natural numbers can appear in mathematical discourse. But the contrary is obvious: Who would ever have used a natural number that has more predecessors than successors?

§ 432 The complete list is not a square

Cantor's diagonalization argument. Ok, I've seen this proof countless times. And like I say it's logically flawed because it requires the a completed list of numerals must be square, which they can' t be. First off you need to understand the numerals are not numbers. They are symbols that represent numbers. Numbers are actually ideas of quantity that represent how many individual things are in a collection. So we aren't working with numbers here at all. We are working with numeral representations of numbers. So look at the properties of our numeral representations of number: Well, to begin with we have the numeral system based on ten. This includes the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. How many different numbers can we list using a column that is a single digit wide? Well, we can only list ten different numbers. 0 1 2 3 4 5 6 7 8 9 Notice that this is a completed list of all possible numbers. Notice also that this list is not square. This list is extremely rectangular. It is far taller than it is wide. Let, apply Cantor's diagonal method to our complete list of numbers that are represented by only one numeral wide. Let cross off the first number on our list which is zero and replace it with any arbitrary number from 1-9 (i.e. any number that is not zero) [...] Ok we struck out zero and we'll arbitrary choose the numeral 7 to replace it. Was the numeral 7 already on our previous list? Sure it was. We weren't able to get to it using a diagonal line because the list is far taller than it is wide. Now you might say, "But who cares? We're going to take this out to infinity!" But that doesn't help at all. [...] We can already see that in a finite situation we are far behind where we need to be, and with every digit we cross off we get exponentially further behind the list. Taking this process out to infinity would be a total disaster. [Divine Insight, Why Cantor's Diagonalization Proof is Flawed (28 June 2012)] http://debatingchristianity.com/forum/viewtopic.php?t=23975 Thanks to Albrecht Storz for the hint to this source

Usually matheologians confuse infinities. A potentially infinits list is always a square up to every n. But the presence of the antidiagonal cannot be excluded. Then they switch infinities. Nothing new. But necessary to mention it over and over again in order to protect newbies from falling into that trap.

§ 433 The reason for calling matheology matheology

Zeitgeist on the odds of my students to understand transfinite set theory:" if they can be convinced that what their senses tell them may be not be the whole picture, then they may have a chance."

Virgil, appearing as Wisely Non-Theist: "There are 'more' real numbers than there are finite definitions to define them with, so most reals can only be defined collectively, not individually."

Ben Bacarrisse emphasized: "They are not 'entire undefinable'. The set of them can be defined." He added: "You can know things about the set. For example, that it can't be bijected with N."

William P. Hughes: "A subcollection of a listable collection may not be listable."

Alan Smaill: "After all, since matheology accepts undefinable real numbers, then why are you trying to suggest that it does not accept undefinable enumerations?"

That is true. I never got a grasp of this idea: Why should undefinable definable enumerations (also known as lists) be exemptet from the list of unlisted exemptions?

§ 434 Cantor und die Axiome / Cantor and the axioms

Cantor hat niemals versucht, seine Mengenlehre axiomatisch zu formalisieren. Erst am Ende seiner aktiven Laufbahn, unter Hilberts Einfluss, hat er Axiome dafür überhaupt in Betracht gezogen. "Er sieht, schon gegen Ende des 19. Jahrhunderts, das Aufkommen eines formalistischen Denkens, das ihm zutiefst zuwider war" (Meschkowski). Die folgenden Paragraphen werden jede mir bekannte Erwähnung von Axiomen in Cantors Werk und Korrespondenz behandeln.

Das sogenannte Cantorsche Axiom (1872) betrifft lediglich die Geometrie:

... ein Axiom hinzuzufügen, welches einfach darin besteht, daß auch umgekehrt zu jeder Zahlengröße ein bestimmter Punkt der Geraden gehört, dessen Koordinate gleich ist jener Zahlengröße ... Ich nenne diesen Satz ein Axiom, weil es in seiner Natur liegt, nicht allgemein beweisbar zu sein. [G. Cantor: "Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen", Math. Annalen 5 (1872) 123 - 132]

Das Archimedische Axiom hielt Cantor für beweisbar und den Euklidschen Satz über den Teil und das Ganze für fragwürdig :

Also ist das sogenannte "Archimedische Axiom" gar kein Axiom, sondern ein, aus dem linearen Größenbegriff mit logischem Zwang folgender Satz. [E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer (1932) p. 409]

{{Ich halte}} an der Ueberzeugung fest, dass das sogenannte "Archimedische Axiom" von mir bewiesen und daß eine Abweichung von diesem "Axiom" eine Verirrung ist. [Cantor an Veronese, 7. Sep. 1890]

Sind nicht eine Menge und die zu ihr gehörige Kardinalzahl ganz verschiedene Dinge? Steht uns nicht erstere als Objekt gegenüber, wogegen letztere ein abstraktes Bild davon in unserm Geiste ist? {{Zurück zu den Wurzeln: Eine Menge ist ein Objekt (der Realität).}} Der alte, so oft wiederholte Satz: "Totum est majus sua parte" darf ohne Beweis nur in bezug auf die, dem Ganzen und dem Teile zugrunde liegenden Entitäten zugestanden werden; dann und nur dann ist er eine unmittelbare Folge aus den Begriffen "totum" und "pars". Leider ist jedoch dieses "Axiom" unzählig oft, ohne jede Begründung und unter Vernachlässigung der notwendigen Distinktion zwischen "Realität" und "Größe" resp. "Zahl" einer Menge, gerade in derjenigen Bedeutung gebraucht worden, in welcher es im allgemeinen falsch wird, sobald es sich um aktual-unendliche Mengen handelt und in welcher es für endliche Mengen nur aus dem Grunde richtig ist, weil man hier imstande ist, es als richtig zu beweisen. [E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer (1932) p. 416f]

Von Hypothesen ist in meinen arithmetischen Untersuchungen über das Endliche und Transfinite überall gar keine Rede, sondern nur von der Begründung des Realen in der Natur Vorhandenen. Sie hingegen glauben nach Art der Metageometer Riemann, Helmholtz und Genossen auch in der Arithmetik Hypothesen aufstellen zu können; was ganz- unmöglich ist; darin liegt Ihre ebenso unglückliche wie verhängnisvolle Täuschung, von welcher ich Sie nicht abbringen kann und mag. So wenig sich in der Arithmetik der endlichen Anzahlen andere Grundgesetze aufstellen lassen, als die seit Alters her an den Zahlen 1, 2, 3, ... erkannten, ebensowenig ist eine Abweichung von den arithmetischen Grundwahrheiten im Gebiete des Transfiniten möglich. "Hypothesen" welche gegen diese Grundwahrheiten verstoßen, sind ebenso falsch und widersprechend, wie etwa der Satz 2 + 2 = 5 oder ein viereckiger Kreis. Es genügt für mich, derartige Hypothesen an die Spitze irgend einer Untersuchung gestellt zu sehen, um von vorn herein zu wissen, daß diese Untersuchng falsch sein muss. Und der Erfolg hat es ja bei Ihnen gezeigt, da Sie durch Ihre beklagenswerthen "Hypothesen" zu dem widersprechenden Begriffe actual unendlich kleiner linearer Größen geführt worden sind! [Cantor an Veronese, 17. Nov. 1890]

Was Herr Veronese darüber in seiner Schrift giebt, halte ich für Phantastereien und was er gegen mich darin vorbringt, ist unbegründet. Ueber seine unendlich großen Zahlen sagt er, daß sie auf anderen Hypothesen aufgebaut seien, als die meinigen. Die meinigen beruhen aber auf gar keinen Hypothesen, sondern sind unmittelbar aus dem natürlichen Mengenbegriff abgezogen; sie sind ebenso nothwendig und frei von WiIlkür, wie die endlichen ganzen Zahlen. [Cantor an Killing, 5. April 1895]

...der Unterschied besteht nur in den "Hypothesen" (Axiomen), die er für das jeweilige System fordert. Mit einer solchen Sichtweise steht Veronese einer modernen Axiomatik deutlich näher als Cantor, der aufgrund seiner philosophischen Ansichten Axiome als "Grundwahrheiten" ansieht. [H. Meschkowski, W. Nilson (Herausgeber): Georg Cantor Briefe , Springer, Berlin (1991) p. 329]

{{Tatsächlich hat Cantor recht, sofern Mathematik als ernsthafte Wissenschaft aufgefasst wird und nicht als frivloes Spiel mit Sinnlosigkeiten.}}

Erst die Erkenntnis inkonsistenter Mengen leitete Cantor zum Axiom der transfiniten Arithmetik:

Die Thatsache der "Consistenz" endlicher Vielheiten ist eine einfache, unbeweisbare Wahrheit, es ist "das Axiom der Arithmetik (im alten Sinne des Wortes)". Und ebenso ist die "Consistenz" der Vielheiten, denen ich die Alephs als Cardinalzahlen zuspreche, "das Axiom der erweiterten, der transfiniten Arithmetik". [Cantor an Dedekind, 28. Aug. 1899]

Daß die "abzählbaren" Vielheiten {αν} fertige Mengen sind, scheint mir ein axiomatisch sicherer Satz zu sein, auf welchem die ganze Functionentheorie beruht. [Cantor an Hilbert, 10. Okt. 1898]

Daß das "arithmetische Continuum" in diesem Sinne eine "Menge" [ist], ist unsere gemeinsame Ueberzeugung; die Frage ist, ob diese Wahrheit eine beweisbare, oder ob sie ein Axiom ist. Ich neige jetzt mehr zu der letzteren Alternative, würde mich aber gerne von Ihnen für die andere überzeugen lassen. [Cantor an Hilbert, 9. Mai 1899]

Unter dem Einfluss Hilberts entsteht schließlich die Cantorsche Axiomatik:

Ich unterscheide in der reinen Mathematik dreierlei Axiome: 1) Die logischen Axiome, die sie mit allen anderen Wissenschaften gemein hat, und die in der formalen Logik, neuerdings im Logikcalcül systematisch behandelt werden. 2) Die physischen Axiome der Mathematik, z. B. die geometrischen Axiome und die Axiome der Mechanik. Sie sind dadurch kenntlich, daß ihnen der Charakter der Nothwendigkeit fehlt, sie können durch andere ersetzt werden (man denke an das Parallelenaxiom Euclid's und die nichteuclidische Geometrie). Ich nenne sie "physische Axiome", weil sie sich auf besondere Naturen beziehen, wie etwa auf Raumdinge, Zeitdinge, Kraftdinge, Massendinge etc. Ausser diesen beiden Arten von Axiomen, existirt noch eine dritte, die bisher unbeachtet geblieben zu sein scheinen, weil sie mehr versteckt sind, als jene. Ich nenne sie die: 3) Metaphysischen Axiome der Mathematik (ich nenne diese Axiome "metaphysisch", weil sie sich auf Dinge überhaupt, gleichviel welche Natur sie haben, beziehen); zu diesen gehören vor Allem die Axiome der Arithmetik, sowohl der endlichen, wie auch der transfiniten Zahlentheorie. Das Axiom der endlichen Zahlentheorie lässt sich kurz so aussprechen: "Jede endliche Vielheit ist consistent." Zur Erläuterung dieses: a) Der Begriff "endliche Vielheit" ist, wie Sie zugeben werden, ohne Heranziehung des Zahlbegriffs bestimmbar. b) Die endlichen Zahlbegriffe resp. Zahlen sind nur unter Voraussetzung der Wahrheit des soeben formulirten Axioms denkbar resp. möglich. Das Axiom der transfiniten Zahlentheorie ist dieses: "Jede Vielheit, zu welcher ein signirtes Alef, ¡γ (wo γ irgend eine Ordnungszahl) gehört, ist consistent". [Cantor an Hilbert, 27. Jan. 1900]

Durch sein untergliedertes Axiomensystem unterscheidet sich Cantor nach eigenen Worten von Dedekind:

Dedekind geht offenbar von der Meinung aus, daß die Zahlentheorie keine anderen Axiome voraussetze als die logischen; dasselbe scheinen die Vertreter des Logikcalcüls zu glauben. In der Vorrede der Dedekindschen Schrift heißt es: die Zahlentheorie "ein Theil der Logik"; die Zahlen sind ihm "freie Schöpfungen des menschlichen Geistes". [Cantor an Hilbert, 27. Jan. 1900]

Doch schon kurz darauf reduziert Cantor sein System wieder rigoros:

Wie ich die Sache ansehe, so sind folgende zwei Axiome als Grundlage unserer endlichen Zahlentheorie nothwendig und hinreichend. I. Es giebt Dinge (d. h. Gegenstände unseres Denkens). II. Ist V eine consistente Vielheit von Dingen und d ein nicht in V als Theil enthaltenes Ding, so ist die Vielheit V + d auch consistent. Diese beiden Axiome liefern mir die unbegrenzte Zahlenreihe 1, 2, 3, 4, ... der endlichen ganzen Cardinalzahlen und alle Gesetze unter ihnen lassen sich beweisen, ohne Zuhülfenahme weiterer Axiome. [Cantor an Hilbert, 20. Feb. 1900]

Eine Fortsetzung der axiomatischen Überlegungen in der anschließenden Korrespondenz mit Hilbert oder anderen Mathematikern ist mir nicht bekannt. Am deutlichsten und zutreffendsten dürfte Cantors persönliche Position aber in einem frühen Brief an Wundt formuliert sein:

Sie haben vollkommen Recht, wenn Sie den Gauß-Riemann-Lobatschewskischen Räumen den realen Untergrund absprechen, dagegen ihre volle Berechtigung als "logische Postulate" zugeben. Dagegen nehme ich für meine unendlichen Zahlenbegriffe in Anspruch, daß sie frei von jeglicher Willkür sich durch Abstraktion aus der Wirklichkeit mit derselben Notwendigkeit ergeben wie die gewöhnlichen ganzen Zahlen, welche bisher allein als Ursprung aller anderen mathematischen Begriffsbildungen gedient haben. Die transfiniten ganzen Zahlen sind keineswegs, wie Sie sagen, bloße "Fiktionen" resp. "logische Postulate", wie es die geometrischen Räume mit n Dimensionen sind, sondern sie haben denselben Charakter der Realität wie die älteren Zahlen: 1, 2, 3 etc. Um dies zu verstehen, setze ich nichts anderes voraus, als eine Weltbetrachtung, für welche die Leibnizschen Worte maßgebend sind: "Je suis lettement pour l'infini actuel, qu'au lieu d'admettre que la nature l'abhorre, comme l'on dit vulgairement, je tiens quelle l'affecte partout, pour mieux marquer les perfections de son Auteur. Ainsi je crois qu'il n'y a aucune partie de la matière qui ne soit, je ne dis pas divisible, mais actuellement divisée" etc. [Cantor an Wundt, 5. Okt. 1883]

"Notwendig und frei von Willkür." An solchen Sachen ist nichts zu deuteln.

Cantor has never attempted to formalize set theory axiomatically. Only toward the end of his active career, under the influence of Hilbert, he has considered axioms for set theory at all. "He sees, already toward the end of the 19th century, the appearance of formalistic thinking that he abhorred" (Meschkowski). The following paragraphs will cover every mentioning, that I am aware of, of axioms in Cantor's work and correspondence.

The so-called Cantor's axiom (1872) concerns geometry only:

... to add an axiom, requiring that, vice versa, to every numerical value there belongs a certain point of the straight line the co-ordinate of which is equal to that numerical value. ... I call this sentence an axiom because it is immanent to its nature that it cannot be proven in general. [G. Cantor: "Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen", Math. Annalen 5 (1872) 123 - 132]

Cantor considered the Archimedian axiom as a provable theorem and the Euclidean proposition about the whole and the part as questionable.

Thus the "Archimedean axiom" is not an axiom but a theorem that follows with logical necessity from the notion of linear magnitude. [E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer (1932) p. 409]

I maintain the position that the so-called "Archimedean axiom" has been proven by myself and that a deviation from this "axiom" means going astray. [Cantor to Veronese, 7 Sep. 1890]

Aren't a set and its corresponding cardinal number quite different things? Does not the first face us as an object whereas the latter is an abstract picture of it in our mind? {{Remember the roots: A set is an object (of reality).}} The old, so often repeated sentence: "The whole is more than its part" may be accepted, without proof, only with respect to the entities which the whole and the part are based upon. Then and only then the sentence is an immediate consequence of the notions "whole" and "part". Unfortunately, however, this "axiom" has been used innumerably often without any grounds and neglecting the necessary distinction between "reality" and "magnitude" just in that meaning which makes it false in general, as soon a actually infinite sets are involved and which makes it right for finite sets only because we are able to prove it as right in this domain. [E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer (1932) p. 416f]

Hypotheses are completely out of the question in my arithmetical investigations about the finite and the transfinite. Only the reasons for the real that is existing in nature are established there. You, on the other hand, believe in the manner of the meta geometers Riemann, Helmholtz, and comrades to be able to establish hypotheses in arithmetic too - which is completely impossible. This is the cause of your as unfortunate as disastrous delusion which I am neither able nor willing to dissuade you from. As little as in the domain of finite arithmetics other fundamental laws can be established than those known from time immemorial for the numbers 1, 2, 3, ..., as little a deviation from the fundamental truths is possible in the domain of the transfinite. "Hypotheses" violating these fundamental truths are as wrong and contradictory as, e.g., the expression 2 + 2 = 5 or a square circle. It is sufficient for me to see such hypotheses being put on top of some investigation in order to know from the outset that this investigation must be wrong. And your success has shown it, since your deplorable "hypotheses" have lead you to the contradictory notion of actually infinitely small linear magnitudes. [Cantor to Veronese, 17 Nov. 1890]

I think the arguing that Mr. Veronese gives in his writings is fantasy, and what I says against me is unreasonable. He says about his infinitely large numbers that they have been based upon other hypotheses than mine. But mine are not based upon hypotheses but have been derived immediately from the notion of set: They are just as necessary and free of arbitrariness as the finite integers. [Cantor to Killing, 5 April 1895]

... The difference consists only in the "hypotheses" (axioms) which he requires for the respective system. With that view Veronese is significantly closer to a modern axiomatics than Cantor who, based upon his philosophical opinions, considers axioms as "fundamental truths". [H. Meschkowski, W. Nilson (Herausgeber): Georg Cantor Briefe , Springer, Berlin (1991) p. 329]

{{In fact Cantor is right if mathematics is perceived as a serious science and not a frivolous play about nonsense.}}

Only the recognition of inconsistent sets led Cantor to the consideration of axioms of transfinite arithmetic.

The fact of the "consistency" of finite multitudes is a simple unprovable truth. It is "the axiom of arithmetic (in the old meaning of the word)". And similarly is the "consistency" of the multitudes to which I attach the alephs "the axiom of the extended, the transfinite arithmetic". [Cantor to Dedekind, 28 Aug. 1899]

That the "countable" multitudes {αν} are completed sets seems to me to be an axiomatically certain theorem which the whole theory of functions rests upon. [Cantor to Hilbert, 10 Oct. 1898]

It is our common conviction that the "arithmetic continuum" in this sense is a "set". The question is whether this truth is a provable one or whether it is an axiom. Currently I tend more towards the latter alternative but I am open to be convinced by you of the other. [Cantor to Hilbert, 9 May 1899]

I distinguish in pure mathematics three kinds of axioms: 1) The logical axioms, that it has in common with all other sciences which in formal logic and recently in the logical calculus have been treated systematically. 2) The physical axioms of mathematics, for instance the geometrical axioms and the axioms of mechanics. They can be recognized by their feature of lacking necessity. They can be replaced by others (think of the parallel axiom of Euclid and and the non-euclidean geometry). I call them physical axioms because they are related to special natures like space-things, time-things, force- things, mass-things, etc. Besides these two kinds of axioms there exists a third kind which hitherto seems to have gone unnoticed because they are more hidden than those. I call them the 3) Metaphysical axioms of mathematics (I call them "metaphysical" because they are related to things in general, no matter what nature they may have). T to this category belong in particular the axioms of arithmetic, both the finite and the infinite numner theory. The axiom of the finite number theory can briefly be noted as: "Every finite multitude is consistent". This as explanation: a) The notion "finite multitude" is, as you will concede, determinable without reference to the notion of number. b) The finite notions of number or numbers themselves are only conceivable or possible when the truth of the above axiom is assumed.

The axiom of the transfinite number theory is this: "Every multitude, which a signed alef, ¡γ (with γ some ) belongs to, is consistent." [Cantor to Hilbert, 27 Jan. 1900]

By his subdivided axiom system Cantor differs, according to his own words, from Dedekind:

Dedkind obviously holds the opinion that number theory assumes no other axioms than the logical ones; the supporters of the logic calculus seem to believe the same. In the preface of Dedekind's paper we read: number theory "a part of logic"; the numbers are for him "free creations of the human spirit". [Cantor to Hilbert, 27 Jan. 1900]

But after a short while already Cantor cuts his axiom system rigorously:

In the way I consider the matter, there are the following two axioms as the foundation of our finite number theory necessary and sufficient. I. There are things (i.e., objects of our thinking). II: If V is a consistent multitude of things and d a thing that is not contained in V, then the multitude V + d is also consistent. These two axioms supply me the unlimited series of numbers 1, 2, 3, 4, ... of the finite integer cardinal numbers, and all laws among them can be proven without using further axioms. [Cantor to Hilbert, 20 Feb. 1900]

I am not aware of any continuation of the axiomatic reflections in the subsequent correspondence with Hilbert or other mathematicians. The most apparent and accurate description of Cantor's personal position, however, might be obtained from an early letter to Wundt:

You are completely right if you deny the real basis of the Gauß-Riemann-Lobachewsky spaces, but accept their full legitimacy as "logical postulates". For my notions of infinite numbers, however, I claim that they result, free of any arbitrariness, from abstraction from the reality with the same necessity as the usual integers which hitherto solely have served as the origin of all other mathematical notions. The transfinite integers are by no means, as you call it, bare "fictions" or "logical postulates" like the geometrical spaces with n dimensions, but they have the same character of reality as the older numbers 1, 2, 3, etc. In order to understand this, I do not assume anything else but a world view for which Leibniz' words set the standard: "I am so much in favour of the actual infinite. I believe that nature, instead of abhorring it, as is usually said, uses it frequently everywhere in order to show better the perfectness of its author. Therefore I believe that there is no piece of matter that not - I don't say is divisible - but actually divided" etc. [Cantor to Wundt, 5 Okt. 1883]

"Necessary and free of arbitrariness." There are no ifs and buts about such stuff.

§ 435 The rising of the empty set

Bernard Bolzano, the inventor of the notion set (Menge) in mathematics would not have named a nothing an empty set. In German this word has the meaning of many. Often we find in German texts the expression große (great or large) Menge, rarely the expression kleine (small) Menge. Therefore Bolzano apologizes for using this word in case of sets having only two elements: Auch einen Inbegriff, der nur zwey Theile enthält, erlaube man mir hier eine Menge zu nennen. (Allow me to call also a collection containing only two parts a set.) [J. Berg (Hrsg.): Bernard Bolzano, Einleitung zur Grössenlehre, Friedrich Frommann Verlag, Stuttgart (1975), Bolzano-Gesamtausgabe, Reihe II Band 7. p. 152].

Also Richard Dedekind discarded the empty set. But he accepted the singleton, i.e., the non- empty set of less than two elements: Für die Gleichförmigkeit der Ausdrucksweise ist es vorteilhaft, auch den besonderen Fall zuzulassen, daß ein System S aus einem einzigen (aus einem und nur einem) Element a besteht, d. h. daß das Ding a Element von S, aber jedes von a verschiedene Ding kein Element von S ist. Dagegen wollen wir das leere System, welches gar kein Element enthält, aus gewissen Gründen hier ganz ausschließen, obwohl es für andere Untersuchungen bequem sein kann, ein solches zu erdichten. (For the uniformity of the wording it is useful to permit also the special case that a system S consists of a single (of one and only one) element a, i.e., that the thing a is elememt of S but every thing different from a is not an element of S. The empty system, however, which does not contain any element shall be excluded completely for certain reasons, although it might be convenient for other investigations to fabricate such. [R. Dedekind: "Was sind und was sollen die Zahlen?" Vieweg, Braunschweig 1887, 8th ed. (1960) p. 2]

Georg Cantor mentioned the empty set with some reservations and only once in all his work: "Es ist ferner zweckmäßig, ein Zeichen zu haben, welches die Abwesenheit von Punkten ausdrückt, wir wählen dazu den Buchstaben O; P ª O bedeutet also, daß die Menge P keinen einzigen Punkt enthält, also streng genommen als solche gar nicht vorhanden ist." (Further it is useful to have a symbol expressing the absence of points. We choose for that sake the letter O. P ª O means that the set P does not contain any single point. So it is, strictly speaking, not existing as such.) [E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer (1932) p. 146]

And even Zermelo who made the Axiom II: Es gibt eine (uneigentliche) Menge, die "Nullmenge" O, welche gar keine Elemente enthält. (Axiom II: There is an (improper) set, the "null-set" O which does not contain any element.) [E. Zermelo: "Untersuchungen über die Grundlagen der Mengenlehre I" Mathematische Annalen 65 (1908) p. 263] Zermelo himself said in private correspondence: It is not a genuine set and was introduced by me only for formal reasons [Zermelo to Fraenkel, 31 March 1921] I increasingly doubt the justifiability of the "null set". Perhaps one can dispense with it by restricting the axiom of separation in a suitable way. Indeed, it serves only the purpose of formal simplification. [Zermelo to Fraenkel, 9 May 1921]

So it is all the more courageous that Zermelo based his number system completely on the empty set: { } = 0, {{ }} = 1, {{{ }}} = 2, and so on. He knew at least that there is only one empty set. But many ways to create the empty set could be devised, like the empty set of numbers, the empty set of bananas, the empty set of unicorns, the uncountably many empty sets of all real singletons, and the empty set of empty sets. Is it the emptiest set? Anyhow, zero means nothing. So we can safely say (pun intended): Nothing is named the empty set.

§ 436 I show with absolute rigour that the cardinality of the second number class (II) is not only different from the cardinality of the first number class but that it is indeed the next higher cardinality; (Ich zeige aufs bestimmteste, daß die Mächtigkeit der zweiten Zahlenklasse (II) nicht nur verschieden ist, von der Mächtigkeit der ersten Zahlenklasse, sondern daß sie auch tatsächlich die nächst höhere Mächtigkeit ist;) [G. Cantor: "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", published by the author himself, Leipzig (1883)]

Two questions come to mind: If one of two statements is false, should the other one be believed? If an author, who was not a matheologian, published such a grave mistake, how would he be called by the matheologians of today?

§ 437 I am convinced that the domain of definable numbers is not finished with the finite magnitudes, (Mit den endlichen Größen ist daher meiner Überzeugung nach der Bereich der definierbaren Größen nicht abgeschlossen,) [G. Cantor: "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", Leipzig (1883)]

Obviously Cantor has not been interested in undefinable numbers. Cantors Theorie umfasst keine (in deutscher oder einer anderen sprechbaren Sprache) undefinierbaren Größen.

§ 438 If I talk about a number in the wider sense, then this happens first in the case that an infinite sequence of rational numbers a1, a2, ..., an, ... is given by a law such that the difference an+m - an with growing n becomes infinitely small, (Wenn ich von einer Zahlengröße im weiteren Sinne rede, so geschieht es zunächst in dem Falle, daß eine durch ein Gesetz gegeben unendliche Reihe von rationalen Zahlen a1, a2, ..., an, ... vorliegt, welche die Beschaffenheit hat, daß die Differenz an+m - an mit wachsendem n unendlich klein wird,) [G. Cantor: "Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen", Math. Annalen 5 (1872) p. 123 - 132]

"Given by a finite law"! Cantor knew that infinite sequences cannot be given in another mode. Cantor never, in his complete oeuvre and correspondence, accepts undefinable numbers, i.e., numbers that cannot be definied in German. He does not even talk about them until 1906 because there was no reason. The set of those fools of matheology who consider or even accept undefinable real numbers and who by this perversion of mind have caused mathematics and mathematicians to become an object of ridicule, honoured not better than the insane inhabitants of a mad house, has been empty at Cantor's times.

§ 439 The process of correctly constructing notions is in my opinion always the same: One takes a thing without properties, that beforehand is nothing but a name or symbol A, and properly endows it with several, even infinitely many understandable predicates, the menaing of which is well known by means of other already existing ideas and which may not contradict each other; (Der Vorgang bei der korrekten Bildung von Begriffen ist m. E. überall derselbe; man setzt ein eigenschaftsloses Ding, das zuerst nichts anderes ist als ein Name oder ein Zeichen A, und gibt demselben ordnungsmäßig verschiedene, selbst unendlich viele verständliche Prädikate, deren Bedeutung an bereits vorhandenen Ideen bekannt ist, und die einander nicht widersprechen dürfen;) [G. Cantor: "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", Leipzig (1883)]

§ 440 The definition of an irrational number always requires a well-defined first-order-infinite set of rational numbers; (Zur Definition einer irrationalen reellen Zahl gehört stets eine wohldefinierte unendliche Menge erster Mächtigkeit von rationalen Zahlen;) [G. Cantor: "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", Leipzig (1883)]

§ 441 Hessenberg's argument

Hessenberg derives the uncountability of the powerset of Ù from the fact that the set S of all natural numbers which are not in their image-sets cannot be enumerated by a natural number n. If this set S is enumerated by n, and if n is not in S, then n belongs to S and must be in S, but then n does not belong to S and so on. [Gerhard Hessenberg; "Grundbegriffe der Mengenlehre", Sonderdruck aus den "Abhandlungen der Fries'schen Schule", I. Band, 4. Heft, Vandenhoeck & Ruprecht, Göttingen (1906) ]

This is a logical paradox of self reference like many others, for instance this: "This set has seven words" is as wrong as its negation: "This set does not have seven words". Such paradoxa have nothing to do with cardinal numbers of sets.

Proof: Enumerate Hessenberg's set S by -1. Since -1 is not a natural number, there is no problem. After having enumerated all subsets of Ù by natural numbers and -1, we can show that the powerset of Ù is countable.

Well this concerns only one paradoxical set. But there may be others. No problem. How many paradoxical sets can be defined in a speakable language like English that is suitable to express all of mathematics (or in any formal language that can be defined in English)? The answer is: At most countably many. Therefore all negative integers are sufficient to enumerate all possible paradoxical sets. Finally it is easy to show that all integers belong to a countable set. No mathematical reason to believe in uncountable sets.

§ 442 If {{the pointset}} P(1) has the cardinality of the second number class (II) [i.e., if P(1) is not countable], then P(1)) can always, but in only in one way, be separated into two sets R and S, (Hat aber P(1) die Mächtigkeit der zweiten Zahlenklasse (II) [d. h. ist P(1)) nicht abzählbar], so läßt sich P(1) stets, und zwar nur auf einzige Weise in zwei Mengen R und S zerlegen,) [G. Cantor: "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", Leipzig (1883)]

Only in one way. That means every point has its own definition. No ambiguity caused by uncountably many undefinable elements. Proven by Cantor with the same strictness as its other proofs.

§ 443 By a "manifold" or "set" I understand in general every multitude which can be understood as a unit, i.e., every embodiment of defined elements which by a law can be connected to become an entity. (Unter einer "Mannigfaltigkeit" oder "Menge" verstehe ich nämlich allgemein jedes Viele, welches sich als Eines denken läßt, d. h. jeden Inbegriff bestimmter Elemente, welcher durch ein Gesetz zu einem Ganzen verbunden werden kann.) [G.Cantor: "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", Leipzig (1883)]

§ 444 A significant difference {{with respect to the related philosophy of Platon, Cusanus, Bruno}} is that I fix, according to the concept, once and for all the different gradings of the proper infinite by number classes (I), (II), (III), and so on, and now consider it a task not only to investigate the relations of the supra-finite numbers mathematically but also, whereever they appear in nature, to substantiate and to follow them. (Ein wesentlicher Unterschied {{zur verwandten Philosophie von Platon, Cusanus, Bruno}} besteht aber darin, daß ich die verschiedenen Abstufungen des Eigentlich-unendlichen durch die Zahlenklassen (I), (II), (III) usw. ein für allemal dem Begriffe nach fixiere und es nun als Aufgabe betrachte, die Beziehungen der überendlichen Zahlen nicht nur mathematisch zu untersuchen, sondern auch allüberall, wo sie in der Natur vorkommen, nachzuweisen und zu verfolgen. [G.Cantor: "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", Leipzig (1883)]

§ 445 Cantor's most famous sentence

By a "set" we mean every gathering together M of certain well-distinguished objects m of our understanding or our thinking (which are called the "elements" of M) into a whole. (Unter einer "Menge" verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten m unsrer Anschauung oder unseres Denkens (welche die "Elemente" von M genannt werden) zu einem Ganzen.) [G. Cantor: "Beiträge zur Begründung der transfiniten Mengenlehre", Math. Annalen 46 (1895) 481-512]

§ 446 Equivalence of sets

The equivalence of sets is the necessary and unmistakable criterion for the equality of their cardinal numbers. [...] If now M ~ N, this is based on a law of assiging, by which M and N are mutually uniquely related to each other; here let the element m of M be related to the element n of N. (Die Äquivalenz von Mengen bildet also das notwendige und untrügliche Kriterium für die Gleichheit ihrer Kardinalzahlen. [...] Ist nun M ~ N, so liegt ein Zuordnungsgesetz zugrunde, durch welches M und N gegenseitig eindeutig aufeinander bezogen sind; dabei entspreche dem Elemente m von M das Element n von N.) [G. Cantor: "Beiträge zur Begründung der transfiniten Mengenlehre", Math. Annalen 46 (1895) 481-512]

Note: Only definable elements can be uniquely related to each other.

§ 447 As "cardinality" or "cardinal number" of M we denote the general notion which, aided by our active intellectual capacity, comes out of the set M by abstracting from the constitution of its different elements m and of the order in which they are given. ("Mächtigkeit" oder "Kardinalzahl" von M nennen wir den Allgemeinbegriff, welcher mit Hilfe unseres aktiven Denkvermögens dadurch aus der Menge M hervorgeht, daß von der Beschaffenheit ihrer verschiedenen Elemente m und von der Ordnung ihres Gegebenseins abstrahiert wird.) [G. Cantor: "Beiträge zur Begründung der transfiniten Mengenlehre", Math. Annalen 46 (1895) 481-512]

Note: Cardinality requires different elements.

§ 448 Cantor obtained not only a scientific delight from his infinite numbers but also an aesthetic pleasure

If I understand the infinite as I happened to do here and in my earlier approaches, then this entails for me a real pleasure, which I greatfully abandon myself to: to observe how the whole notion of number, which in the finite has only the background of (counting) number, when we climb to the infinite, so to speak splits into two notions, in that of cardinality, which is independent of the order, that the set has been given, and the (counting) number, which necessarily depends on a pattern that the set has been given and by which the set becomes a well-ordered set. And when I descend back again from the infinite to the finite, then I see as clearly and beautifully how those two notions re-unite and flow together into the notion of the finite integer. (Fasse ich das Unendliche so auf, wie dies von mir hier und bei meinen früheren Versuchen geschehen ist, so folgt daraus für mich ein wahrer Genuß, dem ich mich dankerfüllt hingebe, zu sehen, wie der ganze Zahlbegriff, der im Endlichen nur den Hintergrund der Anzahl hat, wenn wir aufsteigen zum Unendlichen, sich gewissermaßen spaltet in zwei Begriffe, in denjenigen der Mächtigkeit, welche unabhängig ist von der Ordnung, die einer Menge gegeben wird, und in den der Anzahl, welche notwendig an eine gesetzmäßige Ordnung der Menge gebunden ist, vermöge welcher letztere zu einer wohlgeordneten Menge wird. Und steige ich wieder herab vom Unendlichen zum Endlichen, so sehe ich ebenso klar und schön, wie die beiden Begriffe wieder Eins werden und zusammenfließen zum Begriffe der endlichen ganzen Zahl.) [G.Cantor: "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", Leipzig (1883)]

§ 449 Since every single element m, when its features are left out of account, becomes a "one", the cardinal number M is itself a certain set composed of nothing but ones, which has existence in our mind as intellectual image or projection of the given set M. (Da aus jedem einzelnen Elemente m, wenn man von seiner Beschaffenheit absieht, eine "Eins" wird, so ist die Kardinalzahl M selbst eine bestimmte aus lauter Einsen zusammengesetzte Menge, die als intellektuelles Abbild oder Projektion der gegebenen Menge M in unserm Geiste Existenz hat.) [G. Cantor: "Beiträge zur Begründung der transfiniten Mengenlehre", Math. Annalen 46 (1895) 481-512]

Note: "every single" element can be identified and its "one" can be indexed by an ordinal number. Otherwise the "ones" could not be counted.

§ 450 Now we will show that the transfinite cardinal numbers can be ordered according to their magnitude and that they form in this order a "well-ordered set", like the finite numbers but in an extended sense. (Es soll nun gezeigt werden, daß die transfiniten Kardinalzahlen sich nach ihrer Größe ordnen lassen und in dieser Ordnung wie die endlichen, jedoch in einem erweiterten Sinne eine "wohlgeordnete Menge" bilden.) [G. Cantor: "Beiträge zur Begründung der transfiniten Mengenlehre", Math. Annalen 46 (1895) 481-512]

§ 451 Therefore all sets are "countable" in an extended sense, in particular all "continua". (Alle Mengen sind daher in einem erweiterten Sinne "abzählbar", im besonderen alle "Kontinua".) [ Cantor to Dedekind, 28 July 1899]

Note: When a set is countable in normal or extended sense, then its elements must be in one-to- one correspondence with the elements of another set that defines countability in normal or extended sense. One-to-one requires to distinguish each "one" element.

§ 452 Infinite definitions (that do not happen in finite time) are non-things. If Königs theorem was true, according to which all "finitely definable" numbers form a set of cardinality ¡0, this would imply that the whole continuum was countable, and that is certainly false. („Unendliche Definitionen" (die nicht in endlicher Zeit verlaufen) sind Undinge. Wäre Königs Satz, daß alle "endlich definirbaren" reellen Zahlen einen Inbegriff von der Mächtigkeit ¡0 ausmachen, richtig, so hieße dies, das ganze Zahlencontinuum sei abzählbar, was doch sicherlich falsch ist.) [Cantor to Hilbert, 6 August 1906]

Surely Cantor was wrong only in the sense that he didn't point out that the notion of definability cannot be absolute, but depends upon the language. [B. Tait, FOM, Who was the first to accept undefinable individuals in mathematics? (2009)] http://www.cs.nyu.edu/pipermail/fom/2009-March/013468.html

No, Dr. Tait, definability has an absolute meaning. An object is definable whenever it can appear as an individual in mathematics. It is defined by the framework of its appearance. And if the whole twaddle of formal languages does not help to define definition, then we define it in English or German. Every mathematical item that is definable in English or German is "definable in mathematics" and vice versa.

§ 453 Grades of definition

Some real numbers are extremely well defined: The small natural numbers like 3 or 5 can be grasped at first glance, even in unary representation. A real number is very well defined, if its value (compared with the unit) can be determined without any error, like all rational numbers the representations of which have a complexity that can be handled by humans or computers. A real number is well defined, if its value can be determined with an error as small as desired, i.e., the number can be put in trichotomy with every very well defined rational number. The irrational numbers with definitions that can be handled by humans or computers belong to this class. A real number is more or less defined, if it can be communicated such that a receiver with more or less mathematical knowledge understands more or less the same as the sender. To this class belong results of calculations that only in principle can be finished.

Even Non-numbers can be defined like the greatest prime number or the smallest positive rational number or the reversal of the digit sequence of π or the lifetime of the universe measured in seconds.

Mathematical objects without definitions, however, cannot exist since all mathematical objects by definition have no other form of existence than existence by definition.

§ 454 Equality and the axioms of natural numbers

In a discussion about equality it was claimed that 1 + 1 can be same as 1. This is true of course, as long it remains undefined what equivalence relation is expressed by "being equal".

Consider the expressions 0 + 0 and 0.

With respect to the script they are different. Even the two zeros in 0 + 0 are different, one of them being that one on the left-hand side and the other one being just the "other". We can distinguish the zeros. We could not, if they were identical in all respects.

If we know that both expressions are meant to represent numbers, we know that they are equal with respect to the property of "being numbers" (and not being cars or stars).

With respect to numerical value we cannot know the result unless we know what "+" and "=" are meaning. As soon as we know the foundations of arithmetic, we see that 0 + 0 = 0. (This situation is comparable to having apples cut to pieces in closed boxes. Before opening the boxes, we cannot know in how many pieces the contained apple has been cut.)

With respect to angular diameter sun is as large as moon. With respect to physical diameter sun is much larger than moon. With respect to volume sun is much, much larger than moon.

Conclusion: Before knowing what kind of comparison is meant, we cannot obtain a result.

With respect to the Peano axioms in their truncated version, we see for instance that S(x) = S(y) implies x = y. Here equality is not defined, so the expression is meaningless. If the script is meant, the sequence defined by the axioms could be 0, 0 + 0, 0 + 0 + 0, etc. or 1, 1^1, 1^1^1, etc. Of course we guess somehow that arithmetical equality is meant as soon as numbers get involved. That means, the reader is not only expected to be able to read and to understand written text, but also to decide when two "successors" are equal or different. A reader who is able to recognize the numerical equality or inequality of numbers would know +1 as well and obtain the sequence Ù from the three axioms:

1 œ M n œ M fl n + 1 œ M Ù Œ M

If unable to read this text, the prospective reader should learn to read. If unable to understand the used logic, the prospective reader should learn its basics. If unable to understand the meaning of "+1" and "=", the prospective reader should learn the basics of arithmetic.

Then the reader would be far better off than with the five Peano axioms in their truncated version which do not define the natural numbers unless this definition is taken from elsewhere.

§ 455 Realness of integers according to Cantor

For one we are allowed to consider the integers inasmuch as being really existing as they, based on definitions, occupy a distinct place in our mind, being extremely well distinguished from all other pieces of our thinking, having well defined relations to them and in a certain way are modifying the substance of our mind; allow me to call this kind of reality intrasubjective or immanent reality. Further realness can be attributed to the the numbers, because they must be considered as an expression or an image of events or relations in the external world opposite to the intellect, and since further the different number classes (I), (II), (III), etc. are representing cardinalities which indeed occur in the physical and intellectual nature. This second kind of reality I call the transsubjective or also transient reality of the integers.

Einmal dürfen wir die ganzen Zahlen insofern für wirklich ansehen, als sie auf Grund von Definitionen in unserm Verstande einen ganz bestimmten Platz einnehmen, von allen übrigen Bestandteilen unseres Denkens aufs beste unterschieden werden, zu ihnen in bestimmten Beziehungen stehen und somit die Substanz unseres Geistes in bestimmter Weise modifizieren; es sei mir gestattet, diese Art der Realität unsrer Zahlen ihre intrasubjektive oder immanente Realität zu nennen. Dann kann aber auch den Zahlen insofern Wirklichkeit zugeschrieben werden, als sie für einen Ausdruck oder ein Abbild von Vorgängen und Beziehungen in der dem Intellekt gegenüberstehenden Außenwelt gehalten werden müssen, als ferner die verschiedenen Zahlenklassen (I), (II), (III) u. s. w. Repräsentanten von Mächtigkeiten sind, die in der körperlichen und geistigen Natur tatsächlich vorkommen. Diese zweite Art der Realität nenne ich die transsubjektive oder auch transiente Realität der ganzen Zahlen. [G. Cantor: "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", Leipzig (1883)]

According to this definition, without at least one atom in the brain devoted to the integer or one atom outside representing it, the integer has neither intrinsic nor transient reality. Numbers exceeding the number of 1080 atoms in the universe can have only matheological reality, i.e., irreality.

§ 456 David Hilbert on the infinite

Finally we will remember our original topic and draw the conclusion. On balance the complete result of all our investigations about the infinite is this: The infinite is nowhere realized; it is neither present in nature nor admissible as the foundation of our rational thinking. This is a remarkable harmony between being and thinking.

Zuletzt wollen wir wieder unseres eigentlichen Themas gedenken und über das Unendliche das Fazit aus allen unseren Überlegungen ziehen. Das Gesamtergebnis ist dann: das Unendliche findet sich nirgends realisiert; es ist weder in der Natur vorhanden, noch als Grundlage in unserem verstandesmäßigen Denken zulässig - eine bemerkenswerte Harmonie zwischen Sein und Denken. [David Hilbert, Über das Unendliche, 24. Juni 1925] http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=26816

If not in the foundations (i.e., as an axiom) how else could the infinite enter mathematics?

§ 457 The infinite human mind

If it turns out that the mind is able to define and to distinguish, in a certain sense infinite, i.e., transfinite numbers, then either the words "finite mind" have to be given an extended meaning [...] or the precicate "infinite" has to be granted to the human mind in a certain respect, the latter of which is in my opinions the only right position.

Zeigt es sich aber, daß der Verstand auch in bestimmtem Sinne unendliche, d. i. überendliche Zahlen definieren und voneinander unterscheiden kann, so muß entweder den Worten "endlicher Verstand", eine erweiterte Bedeutung gegeben werden [...]; oder es muß auch dem menschlichen Verstand das Prädikat "unendlich" in gewissen Rücksichten zugestanden werden, was meines Erachtens das einzig Richtige ist. [G. Cantor: "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", Leipzig (1883)]

Alas it has turned out that the mind is unable to define or to distinguish infinitely many numbers. All numbers that ever have been defined as individuals, including real numbers, complex numbers, and more-complex numbers like tensors, functions, and sets, belong to a finite set - and that will remain so forever.

§ 458 Another resolution of Berry's paradox

Quite a lot of resolutions of the Berry paradox have been proposed http://en.wikipedia.org/wiki/Berry_paradox

Here is another one, based upon the different grades of definition (cp. § 453) resulting in distinctions like this: The set { x | x2 - 3x + 2 = 0 } is very well defined. The set {1, 2} is extremely well defined.

Applied to the Berry paradox we find that "the least natural not nameable in fewer than nineteen syllables" has been very well defined only, using 18 syllables, but by this definition it is not immediately clear what number is meant. Some work is required to find a definition that makes this number extremely well defined: "one-hundred and eleven thousand, seven-hundred and seventy seven" or 111,777.

The paradox vanishes as soon as definitions are distinguished by their grade.

§ 459 Every set can be well-ordered

That it is always possible to give every well-defined set the form of a well-ordered set, on this, as it appears to me, fundamental and momentous and by its universality particularly remarkable law of thinking I will come back in a later treatise. Daß es immer möglich ist, jede wohldefinierte Menge in die Form einer wohlgeordneten Menge zu bringen, auf dieses, wie mir scheint, grundlegende und folgenreiche, durch seine Allgemeingültigkeit besonders merkwürdige Denkgesetz werde ich in einer späteren Abhandlung zurückkommen. [G. Cantor: "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", Leipzig (1883)]

Proof that every set can be well-ordered. Beweis, daß jede Menge wohlgeordnet werden kann [E. Zermelo: "Beweis, daß jede Menge wohlgeordnet werden kann", Mathematische Annalen 59 (1904) 514-516] New proof of the possibility of a well-ordering {{with the page header}} New proof of the well- ordering. Neuer Beweis für die Möglichkeit einer Wohlordnung {{mit der Seitenüberschrift}} Neuer Beweis für die Wohlordnung. [Ernst Zermelo: "Neuer Beweis für die Möglichkeit einer Wohlordnung", Mathematische Annalen 65 (1908) 107-128]

Every set can be well-ordered. This has been proved, to his own satisfaction, by G. Cantor, and to the satisfaction of matheologians by E. Zermelo - even twice. These proofs claim definitely, explicitly, and unanimously that it can be done. Meanwhile matheologians keep being satified with these proofs but explain that it only could be done if it could be done or that there exists a well-ordering of every set, i.e., it has been done by their Gods and Goddesses, since probably even a matheolologian would hardly assume that a set can well-order itself.

§ 460 Cantor's criterion for existence and reality of numbers

Mathematics is completely free in its development and only obliged to obey the self-evident condition that its notions are free of internal contradictions and are related by means of fixed definitions to the already existig and well-established notions. In particular when introducing new numbers, mathematics is merely obliged to give definitions of them, by which process they gain such a definitness and possibly such a relation to the older numbers that they definitely can be distinguished from each other. As soon as a number satifies all these critera it may be and has to be considered as existing and having reality in mathematics. Die Mathematik ist in ihrer Entwicklung völlig frei und nur an die selbstredende Rücksicht gebunden, daß ihre Begriffe sowohl in sich widerspruchslos sind, als auch in festen durch Definitionen geordneten Beziehungen zu den vorher gebildeten, bereits vorhandenen und bewährten Begriffen stehen. Im besonderen ist sie bei der Einführung neuer Zahlen nur verpflichtet, Definitionen von ihnen zu geben, durch welche ihnen eine solche Bestimmtheit und unter Umständen eine solche Beziehung zu den älteren Zahlen verliehen wird, daß sie sich in gegebenen Fällen untereinander bestimmt unterscheiden lassen. Sobald eine Zahl allen diesen Bedingungen genügt, kann und muß sie als existent und real in der Mathematik betrachtet werden. [G. Cantor: "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", Leipzig (1883)]

However numbers not satisfying these criteria are accepted in matheology.

§ 461 What should hinder us?

[...] forsooth I would not know what should hinder us in our activity of forming new numbers, as soon as it becomes clear that, for the progress of science, it has become desirable or even indispensable to include one of these infinitely many number-classes into examination. [...] ich wüßte aber fürwahr nicht, was uns von dieser Tätigkeit des Bildens neuer Zahlen zurückhalten sollte, sobald es sich zeigt, daß für den Fortschritt der Wissenschaften die Einführung einer neuen von diesen unzähligen Zahlenklassen in die Betrachtung wünschenswert oder sogar unentbehrlich geworden ist. [G. Cantor: "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", Leipzig (1883)]

I know what hinders us in our activity of forming uncountably many numbers: The lack of definitions, i.e., the lack of names. But if uncountably many are there without having been created, why should we bother to create anything which is necessarily belonging to a vanishing minority?

§ 462 Transformations preserving well-ordering

The question by which transformations of a well-ordered set its number of elements is changed, by which it is not, simply can be answered in this way: Those and only those transformations do not change the number of elements which can be put down to a finite or infinite set of transpositions, i.e., of exchanges of two elements. Die Frage, durch welche Umformungen einer wohlgeordneten Menge ihre Anzahl geändert wird, durch welche nicht, läßt sich einfach so beantworten, daß diejenigen und nur diejenigen Umformungen die Anzahl ungeändert lassen, welche sich zurückführen lassen auf eine endliche oder unendliche Menge von Transpositionen, d. h. von Vertauschungen je zweier Elemente. [E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer (1932) p. 214]

Can an infinite number of transpositions be finished? Only if infinity can be finished at all. But that would allow us to obtain the well-ordered set Ù with the largest natural number as the first element, exchanging, in the natural well-ordering, successively, for every n in Ù, the first number n0 and the number n, whenever n > n0.

§ 463 David Hilbert on Potential and Actual Infinity

Should we briefly characterize the new view of the infinite introduced by Cantor, we could certainly say: In analysis we have to deal only with the infinitely small and the infinitely large as a limit-notion, as something becoming, arising, being under construction, i.e., as we put it, with the potential infinite. But this is not the proper infinite. This we have for instance when we consider the entirety of the numbers 1, 2, 3, 4, ... itself as a completed unit, or the points of a length as an entirety of things which is completely available. This sort of infinity is named actual infinite. Will man in Kürze die neue Auffassung des Unendlichen, der Cantor Eingang verschafft hat, charakterisieren, so könnte man wohl sagen: in der Analysis haben wir es nur mit dem Unendlichkleinen und dem Unendlichengroßen aIs Limesbegriff, als etwas Werdendem, Entstehendem, Erzeugtem, d. h., wie man sagt, mit dem potentiellen Unendlichen zu tun. Aber das eigentlich Unendliche selbst ist dies nicht. Dieses haben wir z. B., wenn wir die Gesamtheit der Zahlen 1, 2, 3, 4, ... selbst als eine fertige Einheit betrachten oder die Punkte einer Strecke als eine Gesamtheit von Dingen ansehen, die fertig vorliegt. Diese Art des Unendlichen wird als aktual unendlich bezeichnet. [David Hilbert: "Über das Unendliche", Mathematische Annalen 95 (1926) p. 167] http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=26816

§ 464 Georg Cantor on Potential and Actual Infinity

In spite of significant differences between the notions of the potential and actual infinite, where the first is a variable finite magnitude, growing above all limits, the latter a constant quantity fixed in itself but beyond all finite magnitudes, it happens deplorably frequently that the one is confused with the other. [...] improper infinite = variable finite = syncategorematice infinitum on the one side and proper infinite = transfinitum = completed infinite = being infinite = categorematice infinitum on the other [...] Trotz wesentlicher Verschiedenheit der Begriffe des potentialen und aktualen Unendlichen, indem ersteres eine veränderliche endliche, über alle Grenzen hinaus wachsende Größe, letzteres ein in sich festes, konstantes, jedoch jenseits aller endlichen Größen liegendes Quantum bedeutet, tritt doch leider nur zu oft der Fall ein, daß das eine mit dem andern verwechselt wird. (p. 374) [...] uneigentlichunendlichem = veränderlichem Endlichem = synkategorematice infinitum einerseits und Eigentlichunendlichem = Transfinitum = Vollendetunendlichem = Unendlichseiendem = kategorematice infinitum andrerseits (p. 391) [E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer (1932)]

§ 465 Bernard Bolzano on Potential and Actual Infinity

[...] a manifold which is larger than every finite one, i.e., a manifold which has the property that every finite set is only part of it, I shall call an infinite manifold. [...] If they, like Hegel, Erdmann, and others, imagine the mathematical infinite only as a magnitude which is variable and only has no limit in its growth (like some mathematicians, as we will see soon, have assumed to explain their notion) so I agree in their reproach of this notion of a magnitude only growing into the infinite but never reaching it. A really infinite magnitude, for instance the length of the line not ending on both sides (i.e. the magnitude of that spatial object containing all points which can be determined by the purely intellectually imagined relation with respect to two points) need not be variable, as indeed it is not in this example. And a magnitude that only can be considered to be larger than considered before and being able of becoming larger than every given (finite) magnitude, may as well permanently remain a finite magnitude only, as in case of each of the numbers 1, 2, 3, 4 .... [...] werde ich eine Vielheit, die grösser als jede endliche ist, d. h. eine Vielheit, die so beschaffen ist, dass jede endliche Menge nur einen Theil von ihr darstellt, eine unendliche Vielheit nennen. [...] Wenn sie, wie Hegel, Erdmann u. A. sich das mathematische Unendliche nur als eine Grösse denken, welche veränderlich ist und in ihrem Wachsthume keine Gränze hat (was freilich manche Mathematiker, wie wir bald sehen werden, als die Erklärung ihres Begriffes aufgestellt haben): so pflichte ich ihnen in ihrem Tadel dieses Begriffes einer in das Unendliche nur wachsenden, nie es erreichenden Grösse selbst bei. Eine wahrhaft unendliche Grösse, z. B. die Länge der ganzen beiderseits gränzenlosen Geraden (d. h. die Grösse desjenigen Raumdinges, das alle Puncte enthält, die durch ihr blosses begrifflich vorstellbares Verhältnis zu zwei gegebenen bestimmt sind), braucht eben nicht veränderlich zu sein, wie sie es denn in dem hier angeführten Beispiele in der That nicht ist; und eine Grösse, die nur stets grösser angenommen werden kann, als wir sie schon angenommen haben, und grösser als jede gegebene (endliche) Grösse zu werden vermag, kann dabei gleichwohl beständig eine bloss endliche Grösse verbleiben, wie dieses namentlich von jeder Zahlengrösse 1, 2, 3, 4 ...... gilt. [Bernard Bolzano: "Paradoxien des Unendlichen", Leipzig (1851) 6f]

§ 466 Ernst Zermelo on Potential and Actual Infinity

In contrast to the notion of natural number the field of analysis needs the existence of infinite sets: "As a consequence, those who are really serious about rejection of the actual infinite in mathematics should stop at general set theory and the lower number theory and do without the whole modern analysis." (1909) Infinite domains "can never be given empirically; they are set ideally and exist only in the sense of a Platonic idea" (1932). In general they can only be defined axiomatically; any inductive or "genetic" way is inadequate. "The infinite is neither physically nor psychologically given to us in the real world, it has to be comprehended and 'set' as an idea in the Platonic sense." (1942) [Heinz.-Dieter Ebbinghaus: "Ernst Zermelo, An Approach to His Life and Work", Springer (2007)]

§ 467 Georg Cantor on Potential and Actual Infinity

There is no further justification necessary when I in the "Grundlagen", just at the beginning, distinguish two notions toto genere different from each other, which I call the improper-infinite and the proper-infinite; they have to be understood as in no way compatible with each other. The frequently, at all times, admitted union or confusion of these two completely disparate notions causes, to my firm conviction, innumerable errors; in particular I see herein the reason why the transfinite numbers have not been discovered before. {{Without this confusion and the chance to exploit the due errors set theory would have become extinct long ago.}} Es bedarf also keiner weiteren Rechtfertigung, daß ich in den "Grundlagen" gleich im Anfang zwei toto genere von einander verschiedene Begriffe unterscheide, welche ich das Uneigentlich- unendliche und das Eigentlich-unendliche nenne; sie müssen als in keiner Weise vereinbar oder verwandt angesehen werden. Die so oft zu allen Zeiten zugelassene Vereinigung oder Vermengung dieser beiden völlig disparaten Begriffe enthält meiner festen Überzeugung nach die Ursache unzähliger Irrtümer; im besonderen sehe ich aber hier den Grund, warum man nicht schon früher die transfiniten Zahlen entdeckt hat. [E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer (1932) p. 395]

§ 468 Richard Dedekind on Potential and Actual Infinity

Everytime when there is a cut (A1, A2) which is not created by a rational number, we create a new, an irrational number. {{This is potential infinity.}} Jedesmal nun, wenn ein Schnitt (A1, A2) vorliegt, welcher nicht durch eine rationale Zahl hervorgebracht wird, so erschaffen wir eine neue, eine irrationale Zahl. [Richard Dedekind: "Stetigkeit und Irrationale Zahlen", Vieweg Braunschweig (1872), 6th edn. (1960) p.13]

There are infinite systems. Proof (a similar reflection can be found in § 13 of the Paradoxien des Unendlichen by Bolzano (Leipzig 1851)) The world of my thoughts, i.e., the collection S of all things which can be object of my thinking, is infinite. For, if s is an element of S, then the thought s' that s can be object of my thinking is itself an object of my thinking. {{This is potential infinity.}} Es gibt unendliche Systeme. Beweis (Eine ähnliche Betrachtung findet sich in § 13 der Paradoxien des Unendlichen von Bolzano (Leipzig 1851)). Meine Gedankenwelt, d. h. die Gesamtheit S aller Dinge, welche Gegenstand meines Denkens sein können, ist unendlich. Denn wenn s ein Element von S bedeutet, so ist der Gedanke s' daß s Gegenstand meines Denkens sein kann, selbst ein Element von S. [Richard Dedekind: "Was sind und was sollen die Zahlen?", Vieweg, Braunschweig (1887), 8th edn. (1960) p. 14]

A system S is called infinite, if it is similar to a proper part of itself; otherwise S is called finite system. [...] S is called infinite if there is a proper part of S into which S can distinctly (similarly) be mapped. {{A complete infinite system S is actual infinity.}} Ein System S heißt unendlich, wenn es einem echten Teile seiner selbst ähnlich ist; im entgegengesetzten Falle heißt S ein endliches System. [...] S heißt unendlich, wenn es einen echten Teil von S gibt, in welchem S sich deutlich (ähnlich) abbilden lässt. [Richard Dedekind: "Was sind und was sollen die Zahlen?", Vieweg, Braunschweig (1887), 8th edn. (1960) p. 13]

§ 469 Aristotle on Potential and Actual Infinity

But the phrase 'potential existence' is ambiguous. When we speak of the potential existence of a statue we mean that there will be an actual statue. It is not so with the infinite. There will not be an actual infinite. The word 'is' has many senses, and we say that the infinite 'is' in the sense in which we say 'it is day' or 'it is the games', because one thing after another is always coming into existence. For of these things too the distinction between potential and actual existence holds. We say that there are Olympic games, both in the sense that they may occur and that they are actually occurring. The infinite exhibits itself in different ways - in time, in the generations of man, and in the division of magnitudes. For generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite, but always different. Again, 'being' has more than one sense, so that we must not regard the infinite as a 'this', such as a man or a horse, but must suppose it to exist in the sense in which we speak of the day or the games as existing things whose being has not come to them like that of a substance, but consists in a process of coming to be or passing away; definite if you like at each stage, yet always different. [...] It is reasonable that there should not be held to be an infinite in respect of addition such as to surpass every magnitude, but that there should be thought to be such an infinite in the direction of division. For the matter and the infinite are contained inside what contains them, while it is the form which contains. It is natural too to suppose that in number there is a limit in the direction of the minimum, and that in the other direction every assigned number is surpassed. In magnitude, on the contrary, every assigned magnitude is surpassed in the direction of smallness, while in the other direction there is no infinite magnitude. [...] But in the direction of largeness it is always possible to think of a larger number: for the number of times a magnitude can be bisected is infinite. Hence this infinite is potential, never actual: the number of parts that can be taken always surpasses any assigned number. But this number is not separable from the process of bisection, and its infinity is not a permanent actuality but consists in a process of coming to be, like time and the number of time. With magnitudes the contrary holds. What is continuous is divided ad infinitum, but there is no infinite in the direction of increase. For the size which it can potentially be, it can also actually be. Hence since no sensible magnitude is infinite, it is impossible to exceed every assigned magnitude; for if it were possible there would be something bigger than the heavens. [...] Our account does not rob the mathematicians of their science, by disproving the actual existence of the infinite in the direction of increase, in the sense of the untraversable. In point of fact they do not need the infinite and do not use it. They postulate only that the finite straight line may be produced as far as they wish. It is possible to have divided in the same ratio as the largest quantity another magnitude of any size you like. Hence, for the purposes of proof, it will make no difference to them to have such an infinite instead, while its existence will be in the sphere of real magnitudes. [Aristotle: "Physics", part 6 - 7] http://www.greektexts.com/library/Aristotle/Physics/eng/1327.html

§ 470 Fraenkel et al. on Potential and Actual Infinity

The language in which one deals with the expressions of a given theory (not with the entities denoted by these expressions!) is called the metalanguage of this theory. In our case the metalanguage will be ordinary English, supplemented by a few symbols and some rules governing their use. The language in which the theory itself is formulated is called object- language of this theory. In our case the object-language is a certain extremely restricted sub- language of ordinary English, again supplemented by a few symbols and their rules. [...] Within the framework of the first-order predicate calculus we have a (potentially) infinite list of individual variables x, y, z, w, x', y', z', w', etc. [p 19f] Dedekind, just like Bolzano four decades before, believed that he had proved the existence of infinite sets. However, not only are their methods incompatible with the restrictions of our axiomatic system but they are just those that lead to the logical antinomies. From the axiomatic viewpoint there is no other way for securing infinite sets {{here actual infinity is meant}} but postulating them, and we shall express an appropriate axiom in several froms. While the first corresponds to Zermelo's original axiom of infinity, the second implicitly refers to von Neumann's method of introducing ordinal numbers. [p. 46] [A.A. Fraenkel, Y. Bar-Hillel, A. Levy: "Foundations of Set Theory", Elsevier (1973)]

§ 471 Thoralf A. Skolem on Potential and Actual Infinity

In order to obtain something absolutely nondenumerable, we would have to have either an absolutely nondenumerably infinite number of axioms or an axiom that could yield an absolutely nondenumerable number of first-order propositions. But this would in all cases lead to a circular introduction of higher infinities; that is, on an axiomatic basis higher infinities exist only in a relative sense. [J. van Heijenoort: "From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931", Harvard University Press, Cambridge, Mass. (1967) p. 296]

§ 472 Gerhard Hessenberg on Potential and Actual Infinity

The present work [...] had originally been scheduled to continue a report which appeared under the title "The infinite in mathematics" in the first volume of this journal. This report was concerned with excluding actually infinite magnitudes from the limit methods, in particular from infinitesimal calculus. The continuation should show that this exclusion does in no way mean to refrain from considering actual infinity in mathematics. On the contrary, the example of nondenumerability of the continuum should show the possibility to distinguish different cardinalities, and Cantor's resulting proof of the existence of transcendental numbers should show the practical importance of this distinction. [...] it has been shown that neither in the elementary chapters of mathematics nor in those denoted by "infinitesimal calculus" a really infinite "magnitude" occurs, that rather the word "infinite" is merely used as an abbreviating description of important facts of the finite. Die vorliegende Arbeit [...] war ursprünglich als Fortsetzung eines unter dem Titel "Das Unendliche in der Mathematik" im ersten Heft dieser Zeitschrift erschienenen Berichtes gedacht, der sich mit der Ausschaltung der aktual unendlichen Größen aus den Grenzmethoden, insbesondere aus der Infinitesimalrechnung beschäftigt. Die Fortsetzung sollte zeigen, daß mit dieser Ausschaltung die Mathematik keineswegs auf die Betrachtung des aktual Unendlichen überhaupt verzichtet. Vielmehr sollte das Beispiel der Nichtabzählbarkeit des Kontinuums die Möglichkeit der Unterscheidung verschiedener unendlicher Mächtigkeiten, und der daraus folgende Cantorsche Beweis der Existenz transzendenter Zahlen die praktische Bedeutung dieser Unterscheidung dartun. [...] dargetan worden, daß weder in den elementaren noch in den als "Infinitesimalrechnung" bezeichneten Kapiteln der Mathematik eine wirklich unendliche "Größe" auftritt; daß vielmehr das Wort "unendlich" lediglich zur abkürzenden Beschreibung wichtiger Tatsachen des endlichen benutzt wird. [Gerhard Hessenberg; "Grundbegriffe der Mengenlehre", Sonderdruck aus den "Abhandlungen der Fries'schen Schule", I. Band, 4. Heft, Vandenhoeck & Ruprecht, Göttingen (1906) Vorwort und § 1]

§ 473 Thomas Jech on Potential and Actual Infinity

Until then, no one envisioned the possibility that infinities come in different sizes, and moreover, mathematicians had no use for "actual infinity". The arguments using infinity, including the Differential Calculus of Newton and Leibniz, do not require the use of infinite sets. [...] Cantor observed that many infinite sets of numbers are countable: the set of all integers, the set of all rational numbers, and also the set of all algebraic numbers. {{This proof is due to Dedekind. Had he observed already that the set of all definable transcendental numbers is countable too (which is a straight-forward extension of his argument), nobody today would talk about set theory.}} Then he gave his ingeneous diagonal argument that proves, by contradiction, that the set of all real numbers is not countable. A consequence of this is that there exists a multitude of transcendental numbers, even though the proof, by contradiction, does not produce a single specific example. {{Small wonder! Up to every digit the anti-diagonal is a rational number. It is impossible to define a transcendental number by its digits.}} [Thomas Jech: "Set Theory", Stanford Encyclopedia of Philosophy (2002)] http://stanford.library.usyd.edu.au/entries/set-theory/

§ 474 Solomon Feferman on Potential and Actual Infinity

The notions of forcing and of generic sets were introduced by Paul Cohen to settle the long- outstanding problems of the logical interrelationships of the axiom of constructibility, the axiom of choice, and the continuum hypothesis, relative to the system of Zermelo-Fraenkel set theory. In this paper we consider extensions of these notions to other contests, namely that of (1st order) number theory and of a part of (2nd order) analysis, and obtain some applications there.These results depend on a general transform lemma concerning forcing [...] By means of this lemma we are also able to obtain some new applications of Cohen's methods in set theory. The most interesting of these are the following: (1) No set-theoretically definable well-ordering of the continuum can be proved to exist from the Zermelo-Fraenkel axioms together with the axiom of choice and the generalized continuum hypothesis. [S. Feferman: "Some applications of the notions of forcing and generic Sets", Talk at the International Symposium on the Theory of Models, Berkeley (1963)] http://matwbn.icm.edu.pl/ksiazki/fm/fm56/fm56129.pdf

Feferman and Levy showed that one cannot prove that there is any non-denumerable set of real numbers which can be well ordered. Moreover, they also showed that the statement that the set of all real numbers is the union of a denumerable set of denumerable sets cannot be refuted. [Abraham A. Fraenkel, Yehoshua Bar-Hillel, Azriel Levy: "Foundations of Set Theory", North Holland, Amsterdam (1973) p. 62]

- I am convinced that the platonism which underlies Cantorian set theory is utterly unsatisfactory as a philosophy of our subject [...] platonism is the medieval metaphysics of mathematics; surely we can do better. - The actual infinite is not required for the mathematics of the physical world. - The question raised in two of the essays of the volume, Is Cantor Necessary?, is answered with a resounding no. [S. Feferman: "In the light of logic", Oxford Univ. Press (1998)]

Feferman zeigt in seinem Aufsatz "Why a little bit goes a long way - Logical foundations of scientifically applicable mathematics" anhand einiger Fallstudien, dass alle gegenwärtig für wissenschaftliche Zwecke erforderliche Mathematik in einem Axiomensystem ausgeführt werden kann, in dem das aktual Unendliche nicht vorkommt. [W. Mückenheim: "Die Geschichte des Unendlichen", 7th edn., Maro, Augsburg, (2012) p. 108]

§ 475 Henri Poincaré on Potential and Actual Infinity

Why do the pragmatists refuse to admit objects that cannot be defined by a limited number of words? Because they are of the opinion that an object does not exist unless it has been thought and that a thought object cannot be comprehended independent of a thinking subject. That is the core of . And for a thinking subject, be it a man or anything similar, hence a finite being, the infinite cannot have any other sense than the possibility to create as many objects as one wishes. But the Cantorians are realists {{a strange use of this word!}} even with respect to the mathematical magnitudes. These magnitudes appear to them as having an independent existence. They do not create geometry, they discover it. Und warum weigern sich die Pragmatiker, Gegenstände zuzulassen, welche nicht durch eine beschränkte Anzahl von Worten festgelegt werden können? Deshalb, weil sie der Ansicht sind, daß ein Objekt nicht existiert, wenn es nicht gedacht ist und daß man ein gedachtes Objekt nicht unabhängig von einem denkenden Subjekt erfassen kann. Das ist der Kernpunkt des Idealismus. Und für ein denkendes Subjekt, sei es nun ein Mensch oder irgendein Wesen, das dem Menschen gleicht, also infolgedessen ein endliches Wesen, kann das Unendliche keinen anderen Sinn haben als die Möglichkeit, so viele Objekte ins Leben zu rufen, als man will. Aber die Cantorianer sind Realisten selbst in bezug auf die mathematischen Größen. Diese Größen scheinen ihnen eine unabhängige Existenz zu besitzen. Sie schaffen die Geometrie nicht, sie entdecken sie. [H. Poincaré: "Letzte Gedanken: Die Mathematik und die Logik", übers. von K. Lichtenecker, Akadademische Verlagsgesellschaft, Leipzig (1913) p. 160f]

§ 476 Detlef Laugwitz on Potential and Actual Infinity

Numbers serve the purposes of counting, mesuring and computing. [...] the logical short circuit that results from transforming the infinitely increasing number of digits of the potential infinite into an actual infinite and identifying ◊2 with the never ending decimal representation 1.4142 ... Zahlen dienen zur Verrichtung des Zählens, des Messens und des Rechnens. [...] den logischen Kurzschluß, der darin liegt, daß man das potentielle Unendlich der unbegrenzt wachsenden Stellenzahl zu einem aktualen Unendlich macht und ◊2 der "nicht abbrechenden Dezimalzahl" 1,4142... gleich setzt. [Detlef Laugwitz: "Zahlen und Kontinuum", BI, Zürich (1986) p. 16] (Quoted from my handwritten notes.)

§ 477 Georg Cantor on Potential and Actual Infinity

To exclude this confusion from the outset I denote the smallest transfinite number by a symbol that differs from the usual symbol ¶ of the improper infinite, namely ω. In fact ω can be considered somehow as a limit which is approached by the variable, finite integer ν, but only in the sense that ω is the smallest transfinite ordinal number, i.e., the smallest fixed and determined number which is is larger than all finite numbers ν; quite like ◊2 is the limit of certain variable, growing rational numbers. Only this is added: The difference between ◊2 and its rational approximations becomes arbitrily small whereas ω - ν is always equal to ω; however, this difference does not change the fact that ω has to be considered as determined and completed as ◊2, and it does not change that ω has as little traces from of the approaching numbers ν as ◊2 has not any traces of its rational approximations. In a certain sense the transfinite numbers themselves are new irrationalities, and indeed the, in my eyes, best method to define the finite irrational numbers, is quite similar to, I would even like to say, in principle quite the same as, the above described method to introduce transfinite numbers. We can strictly say: the transfinite numbers stand or fall with the finite irrational numbers; they are essentially alike in their basic features because these and those are determined distinguished shapes and modifications (αϕωρισμενα) of the actual infinite. Um diese Verwechslung von vornherein auszuschließen, bezeichne ich die kleinste transfinite Zahl mit einem von dem gewöhnlichen, dem Sinne des Uneigentlich-unendlichen entsprechenden Zeichen ¶ verschiedenen Zeichen, nämlich mit ω. Allerdings kann ω gewissermaßen als die Grenze angesehen werden, welcher die veränderliche endliche ganze Zahl ν zustrebt, doch nur in dem Sinne, daß ω die kleinste transfinite Ordnungs-Zahl, d. h. die kleinste festbestimmte Zahl ist, welche größer ist als alle endlichen Zahlen ν; ganz ebenso wie ◊2 die Grenze von gewissen veränderlichen, wachsenden rationalen Zahlen ist, nur daß hier noch dies hinzukommt, daß die Differenz von ◊2 und diesen Näherungsbrüchen beliebig klein wird, wogegen ω - ν immer gleich ω ist; dieser Unterschied ändert aber nichts daran, daß ω als ebenso bestimmt und vollendet anzusehen ist, wie ◊2, und ändert auch nichts daran, daß ω ebensowenig Spuren der ihm zustrebenden Zahlen ν an sich hat, wie ◊2 irgend etwas von den rationalen Näherungsbrüchen. Die transfiniten Zahlen sind in gewissem Sinne selbst neue Irrationalitäten und in der Tat ist die in meinen Augen beste Methode, die endlichen Irrationalzahlen zu definieren, ganz ähnlich, ja ich möchte sogar sagen im Prinzip dieselbe wie meine oben beschriebene Methode der Einführung transfiniter Zahlen. Man kann unbedingt sagen: die transfiniten Zahlen stehen oder fallen mit den endlichen Irrationalzahlen; sie gleichen einander ihrem innersten Wesen nach; denn jene wie diese sind bestimmt abgegrenzte Gestaltungen oder Modifikationen (αϕωρισμενα) des aktualen Unendlichen. [E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer (1932) p. 395f]

In fact the decimal representations of irrational numbers require an actually infinite sequence of digits - more than any finite sequence. But all finite sequences of the rational approximations also form an infinite sequence leaving out none of the digits belonging to the infinite sequence. So, a distinction by digits is not possible. Only the "not ending", a finitely defined feature, can make the difference.

§ 478 Paul Lorenzen on Potential and Actual Infinity

The finite world-models of present natural science clearly show how the power of the idea of actual infinity has come to an end in classical (modern) physics. In this light the inclusion of the actual infinite into mathematics which explicitly started by the end of the last century with G. Cantor appears disconcerting. In the intellectual overall picture of our century - in particular in view of existentialist philosophy - the actual infinite appears as an anachronism. [...] We introduce numbers for counting. This does not at all imply the infinity of numbers. For, in what way should we ever arrive at infinitely-many countable things? [...] (1) Start with I. (2) If x has been reached, add xI. These rules [...] supply a constructive definition of numbers (namely the scheme for construction). Now we can immediately say that according to these rules infinitely many numbers are possible: For every number x there remains to construct xI. We have to observe that here only the possibility is claimed - and this is secured just by the rules themselves. [...] To assert however that infinitely many numbers would really be, i.e., really would have been constructed according to these rules - that would be false of course. [...] In philosophical terminology we say that the infinite of the number sequence is only potential, i.e., existing only as a possibility. [...] In arithmetic - we may be allowed to summarize - there does not exist a motive to introduce the actual infinite. The surprising appearance of actual-infinity in modern mathematics therefore can only be understood by including geometry into consideration. Die endlichen Weltmodelle der gegenwärtigen Naturwissenschaft zeigen deutlich, wie diese Herrschaft eines Gedankens einer aktualen Unendlichkeit mit der klassischen (neuzeitlichen) Physik zu Ende gegangen ist. Befremdlich wirkt dem gegenüber die Einbeziehung des Aktual- Unendlichen in die Mathematik, die explizit erst gegen Ende des vorigen Jahrhunderts mit G. Cantor begann. Im geistigen Gesamtbilde unseres Jahrhunderts - insbesondere bei Berücksichtigung des existentialistischen Philosophierens - wirkt das Aktual-Unendliche geradezu anachronistisch. [...] Wir führen die Zahlen zum Zählen ein. Hieraus folgt keineswegs die Unendlichkeit der Zahlen. Denn auf welche Weise sollten wir jemals zu unendlich-vielen zählbaren Dingen gelangen? [...] (1) Man fange mit I an. (2) Ist man zu x gelangt, so füge man noch xI an. Diese Regeln [...] liefern eine konstruktive Definition der Zahlen (nämlich ihr Konstruktionsschema). Jetzt können wir sofort sagen, daß nach diesen Regeln unendlich viele Zahlen möglich sind: zu jeder Zahl x ist ja noch xI zu konstruieren. Man muß darauf achten, daß hier nur die Möglichkeit behauptet wird - und diese ist gerade durch die Regel selbst gesichert. [...] Dagegen zu behaupten, daß unendlich viele solche Zahlen wirklich seien, also wirklich nach dieser Regel konstruiert seien - das wäre natürlich falsch. [...] In philosophischer Terminologie sagt man, daß das Unendliche der Zahlenfolge nur potentiell, d. h. nur als Möglichkeit existiere. [...] In der Arithmetik - so wird man zusammenfassend sagen können - liegt kein Motiv zur Einführung von Aktual-Unendlichem vor. Das überraschende Auftreten von Aktual-Unendlichem in der modernen Mathematik ist daher nur zu verstehen, wenn man die Geometrie mit in die Betrachtung einbezieht. [P. Lorenzen: "Das Aktual-Unendliche in der Mathematik", Philosophia naturalis 4 (1957) 3-11] http://books.google.de/books?id=K0duwwznAzQC&pg=PA195&hl=de&source=gbs_toc_r&cad= 4#v=onepage&q&f=false

§ 479 Herbert B. Enderton on Potential and Actual Infinity

There was no objection to a "potential infinity" in the form of an unending process, but an "actual infinity" in the form of a completed infinite set was harder to accept. [H.B. Enderton: "Elements of Set Theory", Academic Press, New York (1977) p. 14f] http://www.amazon.de/Elements-Set-Theory-Herbert- Enderton/dp/0122384407#reader_0122384407

§ 480 Edward Nelson on Potential and Actual Infinity

Numerals constitute a potential infinity. Given any numeral, we can construct a new numeral by prefixing it with S. Now imagine this potential infinity to be completed. Imagine the inexhaustible process of constructing numerals somehow to have been finished, and call the result the set of all numbers, denoted by Ù. Thus Ù is thought to be an actual infinity or a completed infinity. This is curious terminology, since the etymology of "infinite" is "not finished". [Edward Nelson: "Hilbert's Mistake" (2007) p. 3] https://web.math.princeton.edu/~nelson/papers/hm.pdf

§ 481 Carl Friedrich Gauß on Potential and Actual Infinity

Concerning [a proof Schumacher's for the angular sum of 180° in triangles with two infinitely long sides] I protest firstly against the use of an infinite magnitude as a completed one, which never has been allowed in mathematics. The infinite is only a mode of speaking, when we in principle talk about limits which are approached by certain ratios as closely as desired whereas others are allowed to grow without reservation. Was nun aber [einen Beweis Schumachers für die Winkelsumme von 180° in Dreiecken mit zwei unendlich langen Seiten] betrifft, so protestiere ich zuvörderst gegen den Gebrauch einer unendlichen Größe als einer Vollendeten, welcher in der Mathematik niemals erlaubt ist. Das Unendliche ist nur eine Facon de parler, indem man eigentlich von Grenzen spricht, denen gewisse Verhältnisse so nahe kommen als man will, während anderen ohne Einschränkung zu wachsen verstattet ist. [Gauß an Schumacher, 12. 7. 1831] http://gdz.sub.uni- goettingen.de/dms/load/img/?PPN=PPN236010751&DMDID=DMDLOG_0068&LOGID=LOG_00 68&PHYSID=PHYS_0222

§ 482 Georg Cantor on Potential and Actual Infinity and Gauß

The erroneous in that piece by Gauss is that he says the completed infinity could not become a subject of mathematical consideration; [...] I am to my great regret unable to refer, with respect to the transfinite numbers and what is connected with them, to an important authority like Gauss, I even find him in this respect among my opponents. Das Irrthümliche in jener Gauss'schen Stelle besteht darin, dass er sagt, das Vollendetunendliche könne nicht Gegenstand mathematischer Betrachtungen werden; [...] ich bin zu meinem grossen Bedauern ausser Stande, mich in Beziehung auf die transfiniten Zahlen und was mit diesen zusammenhängt auf eine so grosse Autorität wie Gauss berufen zu können, finde ihn sogar in dieser Beziehung unter meinen Gegnern. [Cantor an Lipschitz, 19. Nov. 1883]

Quite two years ago Mr. Rudolf Lipschitz of Bonn has lead my attention to a certain part of the correspondence between Gauß and Schumacher, where the former objects to every use of the actual infinite in mathematics (letter of July 12, 1831); I have answered in great detail and have rejected in this point the authority of Gauß, which I hold in high esteem in all other relations. Es sind jetzt gerade zwei Jahre her, daß mich Herr Rudolf Lipschitz in Bonn auf eine gewisse Stelle im Briefwechsel zwischen Gauß und Schumacher aufmerksam machte, wo ersterer gegen jede Heranziehung des Aktual-Unendlichen in die Mathematik sich ausspricht (Brief v. 12. Juli 1831); ich habe ausführlich geantwortet und die Autorität von Gauß, welche ich in allen anderen Beziehungen so hoch halte, in diesem Punkte abgelehnt. [E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer (1932) p. 371]

If however, based upon a justified aversion against such illegitimate actual infinity (A.I.), in wide domains of science under the influence of modern epicurean-materialistic mainstream a certain horror infinity has been developed, which has found its classical expression and foundation in the mentioned letter by Gauß, it appears to me that the herewith connected uncritical rejection of the legitimate A.I. is not less a misdemeanour against the nature of things, which have to be taken as they are. And we cannot but conceive this behaviour as somewhat short-sighted because it robs us of the possibility to see the A.I. although, in its highest manifestation, it has created us and maintains us, and in its secondary transfinite forms it surrounds us all-over and even inhabits our mind. Wenn aber aus einer berechtigten Abneigung gegen solche illegitime A. U. sich in breiten Schichten der Wissenschaft, unter dem Einflusse der modernen epikureisch-materialistischen Zeitrichtung, ein gewisser Horror Infiniti ausgebildet hat, der in dem erwähnten Schreiben von Gauß seinen klassischen Ausdruck und Rückhalt gefunden, so scheint mir die damit verbundene unkritische Ablehnung des legitimen A. U. kein geringeres Vergehen wider die Natur der Dinge zu sein, die man zu nehmen hat, wie sie sind, und es läßt sich dieses Verhalten auch als eine Art Kurzsichtigkeit auffassen, welche die Möglichkeit raubt, das A. U. zu sehen, obwohl es in seinem höchsten, absoluten Träger uns geschaffen hat und erhält und in seinen sekundären, transfiniten Formen uns allüberall umgibt und sogar unserm Geiste selbst innewohnt. [E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer (1932) p. 374f]

[...] it seems that the ancients haven't had any clue of the transfinite, the possibility of which is even strongly rejected by Aristotle and his school as in newer times by d'Alembert Lagrange, Gauß, Cauchy and their adherents. [...] daß die Alten keine Ahnung vom Transfiniten gehabt zu haben scheinen, deßen Möglichkeit sogar von Aristoteles und seiner Schule heftig bestritten wird, wie auch in der neueren Zeit von d'Alembert Lagrange, Gauß, Cauchy und deren Anhängern.[Cantor an Peano, 21. Sep. 1895]

My opposition to Gauss consists in the fact that Gauss rejects as inconsistent (I mean he does so unconsciously, i.e., without knowing this notion) all multitudes with exception of the finite and therefore categorically and basically discards the actual infinite which I call transfinitum, and together with it he declares the transfinite numbers as impossible, the existence of which I have founded. Mein Gegensatz zu Gauss besteht hingegen darin, daß Gauss alle Vielheiten, mit Ausnahme der endlichen, für inconsistent hält (ich meine unbewusst, d. h. ohne den Begriff zu haben) und daher kategorisch und principiell dasjenige Acutalunendliche, welches ich Transfinitum nenne, verwirft, mithin auch die transfiniten Zahlen, deren Existenz ich begründet habe, für unmöglich erklärt [Cantor an Hilbert, 27. Jan. 1900]

These are Cantor's written opinions on what Gauß could have meant when talking about something "which never has been allowed in mathematics". Of course the modern masters of matheology know better and slightly different interpretations. They believe that Gauß would welcome Cantor's theory and would give three cheers. From that we can derive three different points of view. 1) Cantor, having been a contemporary of Gauß for 10 years, knew how people of his times used to express themselves in case they wanted to announce an opinion. Then the masters of matheology themselves must be rather stupid. 2) Modern masters of matheology show that Cantor, with regard to one of the most important facts in his whole life, must have been rather stupid. 3) Modern masters of matheology are not stupid but intelligent enough to recognize that their followers are stupid enough to be deceived by their masters without recognizing it.

§ 483 Ernst Zermelo on Potential and Actual Infinity

But in order to save the existence of "infinite" sets we need yet the following axiom, the contents of which is essentially due to Mr. R. Dedekind. Axiom VII. The domain contains at least one set Z which contains the null-set as an element and has the property that every element a of which corresponds to another one of the form {a} or which with every of its elements a contains also the corresponding set {a} as an element. Um aber die Existenz "unendlicher" Mengen zu sichern, bedürfen wir noch des folgenden, seinem wesentlichen Inhalte von Herrn R. Dedekind herrührenden Axiomes. Axiom VII. Der Bereich enthält mindestens eine Menge Z, welche die Nullmenge als Element enthält und so beschaffen ist, daß jedem ihrer Elemente a ein weiteres Element der Fom {a} entspricht, oder welche mit jedem ihrer Elemente a auch die entsprechende Menge {a} als Element enthält. [E. Zermelo: "Untersuchungen über die Grundlagen der Mengenlehre. I.", Math. Ann. 65 (1908) 261-281, p. 266f]

Axiom VII originally guarantees the existence of a potentially infinite set. That's how it has been devised by Bolzano and Dedekind: I can think that I can think that I can think ... Never will I have completed an infinity of thoughts. Only the necessary interpretation of infinite sets as completed infinities in set theory forces set theorists to erroneously take the axiom in an actual sense. A countable set S is in bijection with the set Ù when no element s œ S and no element n œ Ù are remaining unpaired. In potential infinity always nearly all elements would be remaining - if they existed. Alas that would mean actual infinity.

§ 484 Adolf Abraham Fraenkel and Azriel Levy on Potential and Actual Infinity

The statement limnض 1/n = 0 asserts nothing about infinity (as the ominous sign ¶ seems to suggest) but is just an abbreviation for the sentence: 1/n can be made to approach zero as closely as desired by sufficiently increasing the integer n. In contrast herewith the set of all integers is infinite (infinitely comprehensive) in a sense which is "actual" (proper) and not "potential". (It would, however, be a fundamental mistake to deem this set infinite because the integers 1, 2, 3, ..., n, ... increase infinitely, or better, indefinitely.) [p. 6] While the preceding explanations make it obvious that the attacks of various philosophers upon the concept of (transfinite) cardinals are unsubstantiated, the attitude of the (neo-) intuitionists that there do not exist altogether non-equivalent infinite sets is consistent, though almost suicidal for mathematics. [p. 62] [A. Fraenkel, A. Levy: "Abstract Set Theory", North-Holland, Amsterdam (1976)] (Quoted from my handwritten notes.)

§ 485 Henri Poincaré on Potential and Actual Infinity

We'll have to state that the mathematicians in considering the notion of infinity tend toward two different directions. For the one the infinite flows out of the finite, for them there exists infinity only because there is an unlimited number of limited possible things. For the others the infinite exists prior to the finite, the finite constituing a small sector of the infinite. Wir werden zunächst feststellen, daß die Mathematiker in der Art, wie sie den Unendlichkeitsbegriff auffassen, zwei entgegengesetzten Richtungen zuneigen. Für die einen fließt das Unendliche aus dem Endlichen, für sie gibt es eine Unendlichkeit, weil es eine unbegrenzte Zahl begrenzter möglicher Dinge gibt. Für die anderen besteht das Unendliche vor dem Endlichen, indem das Endliche sich als ein kleiner Ausschnitt aus dem Unendlichen darstellt. [H. Poincaré: "Letzte Gedanken: Die Mathematik und die Logik", übers. von K. Lichtenecker, Akadademische Verlagsgesellschaft, Leipzig (1913) p. 145] https://archive.org/stream/letztegedanken00lichgoog#page/n162/mode/2up

§ 486 Georg Cantor on Potential and Actual Infinity

{{It must be hard for English-speaking readers to understand Cantor's lengthy and interlocking sentences. Instead of shortening them, I have inserted parentheses like in mathematics.}} To the idea (to consider the infinite (not only in form of the unlimited growing and the closely connected form of the convergent infinite series (introduced first in the seventeenth century) but also to fix it by numbers in the definite form of the completed-infinite)) I have been forced logically (nearly against my own will because in opposition to highly esteemed tradition) by the development of many years of scientific efforts and attempts, and therefore I do not believe that reasons could be raised which I would not be able to answer. Zu dem Gedanken, das Unendlichgroße nicht bloß in der Form des unbegrenzt Wachsenden und in der hiermit eng zusammenhängenden Form der im siebzehnten Jahrhundert zuerst eingeführten konvergenten unendlichen Reihen zu betrachten, sondern es auch in der bestimmten Form des Vollendet-unendlichen mathematisch durch Zahlen zu fixieren, bin ich fast wider meinen Willen, weil im Gegensatz zu mir wertgewordenen Traditionen, durch den Verlauf vieljähriger wissenschaftlicher Bemühungen und Versuche logisch gezwungen worden, und ich glaube daher auch nicht, daß Gründe sich dagegen werden geltend machen lassen, denen ich nicht zu begegnen wüßte. [E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 175]

§ 487 I never met a mathematician who to a higher degree than Hermite has been a realist in the sense of Plato, and yet I can claim that I never met a more decided opponent of the Cantorian ideas. This is the more a seeming contradiction, as he himself stated frankly: I am an opponent of Cantor because I am a realist. Niemals bin ich einem Mathematiker begegnet, der in höherem Maße ein Realist im Sinne Platos war als Hermite und doch kann ich behaupten, daß ich keinem entschiedeneren Gegner der Cantorschen Richtung begegnet bin. Es ist das ein scheinbarer Widerspruch, um so mehr, als er selbst aus freien Stücken erklärt: Ich bin ein Gegner Cantors, weil ich ein Realist bin. [H. Poincaré: "Letzte Gedanken: Die Mathematik und die Logik", übers. von K. Lichtenecker, Akadademische Verlagsgesellschaft, Leipzig (1913) p. 162f] https://archive.org/stream/letztegedanken00lichgoog#page/n180/mode/2up

§ 488 Couldn't just this seemingly so fruitful hypothesis of the infinite have straightly inserted contradictions into mathematics and have fundamentally distroyed the basic nature of this science which is so proud on its consistency? Könnte nicht gerade diese scheinbar so fruchtbare Hypothese des Unendlichen geradezu Widersprüche in die Mathematik hineingebracht und damit das eigentliche Wesen dieser auf ihre Folgerichtigkeit so stolzen Wissenschaft von Grund auf zerstört haben? [E. Zermelo: "On the hypothesis of the infinite" Warsaw notes W4, quoted in Heinz-Dieter Ebbinghaus: "Ernst Zermelo, An Approach to His Life and Work", Springer (2007) p. 292]

§ 489 Russell would certainly reply that not psychology but logic and are concerned, and then I would be tempted to answer that neither logic nor epistemology are independent of psychology. And this declaration would certainly conclude the argument because it would picture clearly an unbridgeable difference of opinion. Russell würde mir sicher entgegenhalten, daß es sich nicht um Psychologie, sondern um Logik und Erkenntnistheorie handelt und ich würde dann dazu geführt werden, zu antworten, daß weder Logik noch Erkenntnistheorie von der Psychologie unabhängig sind, und dieses Bekenntnis würde wohl die Auseinandersetzung beschließen, weil es eine unüberbrückbare Verschiedenheit der Auffassung zutage fördern würde." [H. Poincaré: "Letzte Gedanken: Die Logik des Unendlichen", übers. von K. Lichtenecker, Akadademische Verlagsgesellschaft, Leipzig (1913) p. 142f] https://archive.org/stream/letztegedanken00lichgoog#page/n160/mode/2up

§ 490 People have asked me, "How can you, a nominalist, do work in set theory and logic, which are theories about things you do not believe in?" ... I believe there is value in fairy tales and in the study of fairy tales. [A.B. Feferman, S. Feferman: "Alfred Tarski - Life and Logic", Cambridge Univ. Press (2004), p. 52]

§ 491 [...] further every mathematical notion carries the necessary corrective in itself; if the notion is unproductive and ineffective, then this will soon become obvious by its uselessness, and it will be abolished because of lack of success. {{This prediction has proven itself wrong in case of transfinite numbers.}} [...] dann aber trägt auch jeder mathematische Begriff das nötige Korrektiv in sich selbst einher; ist er unfruchtbar und unzweckmäßig, so zeigt er es sehr bald durch seine Unbrauchbarkeit und er wird dann wegen mangelnden Erfolges fallen gelassen. [E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 182]

§ 492 I must regard a theory which refers to an infinite totality as meaningless in the sense that its terms and sentences cannot posses the direct interpretation in an actual structure that we should expect them to have by analogy with concrete (e.g., empirical) situations. This is not to say that such a theory is pointless or devoid of significance. {{Let alone to say the contrary.}} [A. Robinson: "Formalism 64" in W.A.J. Luxemburg, S. Koerner (eds.): "A. Robinson: Selected Papers", North Holland, Amsterdam (1979)]

§ 493 The limit process has won, since the limit is an unavoidable notion the importance of which is not touched by the assumption or rejection of the infinitely small. But once it is accepted we see clearly that it makes the infinitely small superfluous. Der Grenzprozeß trug den Sieg davon; denn der Limes ist ein unvermeidlicher Begriff, dessen Wichtigkeit von der Annahme oder Verwerfung des Unendlichkleinen nicht berührt wird. Hat man ihn aber einmal gefaßt, so sieht man, daß er das Unendlichkleine überflüssig macht. [Hermann Weyl: "Philosophie der Mathematik und der Naturwissenschaft", 7. Aufl., Oldenbourg, München (2000) p. 64]

§ 494 [...] as such main issues I mention here the sharp separation of the finite from the infinite, the notion of number of things, the proof that the proof-method known by the name of complete induction (or the conclusion from n on n + 1) is really evidential, and that also the definition by induction (or recursion) is definite and free of contradictions. [...] als solche Hauptpunkte erwähne ich hier die scharfe Unterscheidung des Endlichen vom Unendlichen, den begriff der Anzahl von Dingen, den Nachweis, daß die unter dem Namen der vollständigen Induktion (oder des Schlusses von n auf n + 1) bekannte Beweisart wirklich beweiskräftig, und daß auch die Definition durch Induktion (oder Rekursion) bestimmt und widerspruchsfrei ist. [Richard Dedekind: "Was sind und was sollen die Zahlen?", Vieweg, Braunschweig (1887) preface.]

§ 495 To the reviewer it seems unfortunate that classical set theory is developed in a separate book so that all scruples - or almost all of them - are reserved for the second volume. This might have the effect that most readers of this present volume will probably not become acquainted with the criticisms at all. It is true that some hints to such scruples are given, but most students might not think that they are important. On the other hand, it must be conceded that the lack of knowledge of the results of foundational research will not mean much to mathematicians who are not especially interested in the logical development of mathematics. [T. Skolem: "Review of: A. A. Fraenkel: Abstract Set Theory. Amsterdam & Groningen, North-Holland Publishing Company, 1953. XII + 479 pp." Mathematica Skandinavica 1 (1953) 313.] http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=179577

§ 496 I cannot consider the set of positive integers as given, for the concept of the actual infinite strikes me as insufficiently natural to consider it by itself. [Luzin to Kuratowski, reported by N.Y. Vilenkin: "In search of infinity", Birkhäuser, Boston (1995) p. 126] http://yakovenko.files.wordpress.com/2011/11/vilenkin1.pdf

§ 497 Why are the rationals countable? Because there exists a very simple method to construct all of them. Take two integers. Construct all combinations and enumerate them. Done. (Infinitely many occur infinitely often. No problem.) Why were the real numbers believend to be uncountable? Because there exists no simple method of construction. The reals were thought to be defined by infinite strings of digits. Cantor had invented that method (again). Why are the real numbers countable? Because they are not defineable by infinite strings of digits but only by finite strings of letters like ◊7. These can be expressed in a very simple way, namely by bits 0 and 1. Construct all finite strings of bits. Enumerate them. Done. For every string that does not yet point to a real number like ◊7, we can find a real number which this string points to like ◊8. (infinitely many occur infinitely often. No problem.) Why does the diagonal argument fail? The list of finite definitions has no diagonal. The list of defined real numbers has an infinite diagonal. But infinite diagonals like infinite strings of digits do not define anything. Every real number defines an infinite string. No infinite string defines a real number (because there is no last digit, and every other digits is not sufficient). So only a finitely defined list can yield a diagonal. But all finite definitions are in the list. Contradiction.

§ 498 From my French friends I heard that the tendency towards super-abstract generalizations is their traditional national trait. I do not entirely disagree that this might be a question of a hereditary disease [...] [V.I. Arnold: "On teaching mathematics" (1997), Transl. A.V. Goryunov] http://pauli.uni-muenster.de/~munsteg/arnold.html

§ 499 What is a real number? A real number is an algorithm that supplies an infinite sequence of digits (or bits etc.). This satisfies the axiom of trichotomy with respect to every rational number. A general pointer to a real number is a finite expression which has been related to the real number in at least one physical system of the universe, usually a dictionary or a text book. A special pointer to a real number contains the space-time coordinates or other identification properties of actual constructions of infinite sequences of digits (or bits etc.). All algorithms and all pointers belong as elements to the countable set of all finite expressions. Therefore any uncountability of the real numbers can be excluded.

§ 500 More than any other question the infinite always has moved so deeply the human soul. The infinite rather like no other idea has effected so inspiringly and fruitfully on the human mind. But more than any other notion the infinite is in need of elucidation. Das Unendliche hat wie keine andere Frage von jeher so tief das Gemüt der Menschen bewegt; das Unendliche hat wie kaum eine andere Idee auf den Verstand so anregend und fruchtbar gewirkt; das Unendliche ist aber auch wie kein anderer Begriff so der Aufklärung bedürftig. [D. Hilbert: "Über das Unendliche", Mathematische Annalen 95 (1925) p. 163]

§ 501 No punishment, within legal boundaries, would be too severe for you for your wrongdoings. [anonymous, "What is a real number", sci.math, 9 May 2014] https://groups.google.com/forum/#!original/sci.math/-GSsWLUKmyo/sldkzJaw9ekJ Unfortunately I don't know this gentleman. But I will be proud to quote his sentence whenever teaching my findings with respect to actual transfinity.

§ 502 Modern axiomatic systems unfortunately entail various consequences which are grossly deviating from reality. Therefore a mathematics connected to reality should be based upon those foundations from which it has originally emerged, namely from counting units and drawing lines. Since mathematics owes its creation to abstraction from observations of reality, a statement like "I + I = II" need not be derived by a proof extending over many pages. This statement is much better a natural foundation of arithmetic than any axiom devised for this purpose. It can be proven more compelling by means of an abacus than by any chain of logical conclusions, and be they even most well-grounded. Da moderne Axiomensysteme leider zu mancherlei wirklichkeitsfernen Konsequenzen führen, sollte eine wirklichkeitsnahe Mathematik aus den Grundlagen entwickelt werden, aus denen sie tatsächlich entstanden ist, nämlich aus dem Zählen von Einheiten und dem Zeichnen von Linien. Denn die Mathematik verdankt ihre Entstehung der Abstraktion aus Beobachtungen der Wirklichkeit. Eine Aussage wie "I + I = II" muss nicht aus einem über viele Seiten sich hinziehenden Beweise hergeleitet werden. Diese Aussage selbst ist eine viel natürlichere Grundlage der Arithmetik als irgendein dazu erdachtes Axiom. Sie kann mit einem Abakus zwingender bewiesen werden als durch jede noch so tiefgründige Kette von logischen Schlüssen. [W. Mückenheim: "Mathematik für die ersten Semester", 3. Aufl, Oldenbourg, München (2011) Vorwort]

§ 503 [...] gradually and unwittingly mathematicians began to introduce concepts that had little or no direct physical meaning. [...] after about 1850, the view that mathematics can introduce and deal with [...] concepts and theories that do not have immediate physical interpretation [...] gained acceptance. [M. Kline: "Mathematical Thought from Ancient to Modern Times", Oxford University Press (1972) p. 129ff]

§ 504 Cantor's Theory, if taken seriously, would lead us to believe that while the collection of all objects in the world of computation is a countable set, and while the collection of all identifiable abstractions derived from the world of computation is a countable set, there nevertheless "exist" uncountable sets, implying (again, according to Cantor's logic) the "existence" of a super-infinite fantasy world having no connection to the underlying reality of mathematics. [David Petry, "Objections to Cantor's Theory", sci.math, sci.logic, 20 Juli 2005] http://groups.google.com/group/sci.logic/msg/02ee220b035488f9?dmode=source {{By the way, Cantor never accepted undefinable numbers, and in fact his diagonal argument does not use or provide them. Only matheologians trying at any cost to save transfinity have devised this unmathematical notion.}}

§ 505 As far as I am concerned, I would propose to adhere to the following rules: 1. Never consider other objects than those which can be defined by a finite number of words. 2. Never forget that every proposition about the infinite is only a substitution, an abbreviated expression of a proposition about the finite. 3. Avoid classifications and definitions which are not well-defined. Was mich anlangt, so würde ich vorschlagen, an den folgenden Regeln festzuhalten: 1. Niemals andere Objekte der Betrachtung zu unterziehen, als solche, die sich durch eine endliche Zahl von Worten definieren lassen. 2. Niemals aus den Augen zu verlieren, daß jede Aussage über das Unendliche nur eine Übertragung, ein gekürzter Ausdruck für eine Aussage über das Endliche ist. 3. Klassifikationen und Definitionen, die nicht wohlbestimmt sind, zu vermeiden. [H. Poincaré: "Letzte Gedanken: Die Logik des Unendlichen", übers. von K. Lichtenecker, Akadademische Verlagsgesellschaft, Leipzig (1913) p. 141f] https://archive.org/stream/letztegedanken00lichgoog#page/n158/mode/2up

§ 506 Sequences generated by algorithms can be specified by those algorithms, but what possibly could it mean to discuss a "sequence" which is not generated by such a finite rule? Such an object would contain an "infinite amount" of information, and there are no concrete examples of such things in the known universe. This is metaphysics masquerading as mathematics. [N.J. Wildberger: "Set Theory: Should You Believe?" (2005)] http://web.maths.unsw.edu.au/~norman/views2.htm

§ 507 I understand free variables, for me they are places for substituting individuals. But I do not understand quantifiers since they refer often to actually infinite universes of abstract objects and I do not believe in the existence of such universes. [J. Slupecki, private communication reported in Jan Mycielski: "On the tension between Tarski’s and his model theory (definitions for a mathematical model of knowledge)", Annals of Pure and Applied Logic 126 (2004) 215-224]

§ 508 David Hilbert in 1904 [...] wrote that sets are thought-objects which can be imagined prior to their elements. [At request of he referee {{praised be the referee!}} who asked what is a thought-object let me add: I understand it to be a thought about an object which may exist or not. Thus it is an electrochemical event in the brain or/and its record in the memory. In particular it is a physical thing in space time {{how many can exist in an infinite eternal universe?}}. Of course it is difficult to characterise any physical phenomena. But we have the ability to recognize thoughts as identical or different, just as we have the ability to recognize a silent lightning from a thunderous one. Hence I understand Hilbert's words as follows: mathematicians {{the modern word is matheologians (because of the close connection with theologians in the belief of the non- existing or, at least, the not detectable}} imagine sets which do not exist, but their thoughts about sets do exist and they can arise prior to the thoughts of most elements in those sets. Moreover, in 1923 he described to some extent the algorithm creating those thoughts {{and he concluded: "Finally we will remember our original topic and draw the conclusion. On balance the complete result of all our investigations about the infinite is this: The infinite is nowhere realized; it is neither present in nature nor admissible as the foundation of our rational thinking. This is a remarkable harmony between being and thinking."}}] [J. Mycielski: "Russel'sparadox and Hilbert's (much forgotten) view of set theory" in "One hundred years of Russell's paradox: mathematics, logic, philosophy" G. Link (ed.), de Gruyter (2004) p. 534] http://books.google.de/books?id=Xg6QpedPpcsC&pg=PA533&lpg=PA533&dq=Russell%27s+P aradox+and+++++Hilbert%27s+(much+forgotten)+View+of+Set+Theory&redir_esc=y#v=onepag e&q=Russell's%20Paradox%20and%20%20%20%20%20Hilbert's%20(much%20forgotten)%20 View%20of%20Set%20Theory&f=false I am very indebted to Fred Jeffries for pointing to this source.

§ 509 I am convinced that this theory on day will belong to the common property of objective science and will be confirmed in particular by that theology which is based upon the holy bible, tradition and the natural disposition of the human race - these three necessarily being in harmony with each other. Darum bin i. auch fest überzeugt, daß diese Lehre dereinst Gemeingut der objectiv-gerichteten Wissenschaft werden u. im Besonderen von derjenigen Theologie bestätiget werden wird, welche auf d. heil. Schrift, Tradition und auf d. natürliche Beanlagung d. menschl. Geschlechts sich gründet, welche drei in nothwendiger Harmonie zu einand. stehen. [Georg Cantor an P. Ignatius Jeiler, 20. Mai 1888]

§ 510 When in the age of the scientific revolution Aristotelian metaphysics became the target of modernizers like Galileo, logic was considered part and parcel of metaphysics and was dismissed together with the philosophy Galileo fought against. For him, formalization of logic was obsolete; what was needed from logic he considered as natural and no real subject of study, certainly no precondition for founding the new science of physics. {{Has this situation changed much? It rather has become even worse. Modern logicians "prove" that things can be done which provably cannot be done. The nimbus of mathematical precision and proof has been undermined and besmirched.}} [G. Link: "Introduction" to "One hundred years of Russell's paradox: mathematics, logic, philosophy" G. Link (ed.), de Gruyter (2004) p. 4]

§ 511 I think I have proved above, and it will become plain enough in the course of this paper, that just as well determined countings as with finite sets can be perfomed with infinite sets, provided that the sets are given a determined law according to which they become well-ordered sets. Ich glaube aber oben bewiesen zu haben und es wird sich dies im folgenden dieser Arbeit noch deutlicher zeigen, daß ebenso bestimmte Zählungen wie an endlichen auch an unendlichen Mengen vorgenommen werden können, vorausgesetzt, daß man den Mengen ein bestimmtes Gesetz gibt, wonach sie zu wohlgeordneten Mengen werden. [Ernst Zermelo (Hrsg.): "Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Mit erläuternden Anmerkungen sowie mit Ergänzungen aus dem Briefwechsel Cantor - Dedekind. Nebst einem Lebenslauf Cantors von Adolf Fraenkel." Georg Olms Verlagsbuchhandlung, Hildesheim (1966) p. 174]

§ 512 Concerning the second tranfinite cardinal number ¡1, I am not completely convinced of its existence. We reach it by considering the collection of ordinal numbers of cardinality ¡0; it is clear that this collection must have a higher cardinality. But the question is whether it is closed, that is whether we may talk about its cardinality without contradiction. In any case an actually infinite can be excluded. Was nun die zweite transfinite Kardinalzahl ¡1 betrifft, so bin ich nicht ganz überzeugt, daß sie existiert. Man gelangt zu ihr durch Betrachtung der Gesamtheit der Ordnungszahlen von der Mächtigkeit ¡0; es ist klar, daß diese Gesamtheit von höherer Mächtigkeit sein muß. Es fragt sich aber, ob sie abgeschlossen ist, ob wir also von ihrer Mächtigkeit ohne Widerspruch sprechen dürfen. Ein aktual Unendliches gibt es jedenfalls nicht. [Henri Poincaré: "Über transfinite Zahlen", Sechs Vorträge über ausgewählte Gegenstände aus der reinen Mathematik und Mathematischen Physik, auf Einladung der Wolfskehl-Kommission der Königlichen Gesellschaft der Wissenschaften gehalten zu Göttingen vom 22. - 28. April 1909, Teubner, Leipzig (1910) p. 48]

§ 513 What is your opinion? A) Do you accept uncountable alphabets, i.e., lists defining the shape and pronounciation of letters and determining their place in the order of all letters? B) Do you accept undefinable real numbers, i.e., real numbers without an algorithm to determine the infinite decimal representation? C) Do you accept (A) and (B)? D) Do you accept neither (A) nor (B)?

§ 514 A survey

In analysis we have sequences (sn) of real numbers sn with (improper) limit limnض sn = ¶. Every integer part [ sn ] of a positive real number can be expressed by a sum of units like [ 50/7 ] = 1 + 1 + 1 + 1 + 1 + 1 + 1

Question A) Do you think that it is possible to have [ limnض sn ] = 0 whereas limnض [ sn ] = ¶ = 1 + 1 + 1 + ... ?

In set theory we have sequences (sn) of sets sn with cardinality | sn |. The cardinality is a measure for the number of elements. Contrary to analysis, infinity, called ¡0, is considered to be the number of really existing elements in an infinite set like Ù. So every element of sn contributes one unit to | sn |.

Question B) Do you think that it is possible to have | limnض sn | = 0 whereas limnض | sn | = ¡0 ?

§ 515 Das Paradoxon des Tristram Shandy vereinfacht und verständlich erklärt / The paradox of Tristram Shandy simplified and made intelligible

Donald Duck wird nie so reich wie sein Onkel Dagobert. Wenn er etwas Geld erhält, dann gibt er es sofort wieder aus, bis auf einen Notdollar, den er immer behält,

Er startet mit zwei Dollars, gibt einen aus, erhält einen anderen, gibt einen aus, usw. Seine Dollarnoten nummeriert er mit Filzstift, damit er immer den ältesten ausgibt. Die folgende Tabelle zeigt die schrittweise Entwicklung der Nummern:

1, 2 2 2, 3 3 3, 4 4 ...

Da er als Comic-Figur unsterblich ist, kann die Mengenlehre zur Berechnung des Grenzwertes seines Kapitals herangezogen werden:

Der mengentheoretische Grenzwert zeigt an, welche Zahlen Donald ewig besitzt. Das ist die leere Menge.

Die mengentheoretische Interpretation dieses Grenzwertes sagt aus, dass Donald alle natürlichen Zahlen ausgibt, die Menge der nicht ausgegeben also leer ist. (Diese Interpretation ist z. B. für die Nummerierung aller rationalen Zahlen und die Nummerierung aller Zeilen einer Cantor-Liste erforderlich. Die Menge der nicht nummerierten Zahlen bzw. Zeilen muss leer sein.)

Der tatsächliche Sachverhalt zeigt aber, dass Donald stets eine Zahl behält, die Menge der nicht ausgegeben also nicht und niemals leer ist. Auch der Grenzwert kann nicht kleiner als 1 $ sein.

Die Lösung dieses offensichtlichen Widerspruchs ist folgende: Die Mengenlehre beweist für jede Zahl n, dass die Menge der bis zur Zahl n nicht ausgegeben Zahlen leer ist. Ebenso gilt aber auch für jede Zahl n, dass sie zu einem endlichen Anfangsabschnitt gehört und dass fast alle natürlichen Zahlen auf sie folgen. Damit ist der Widerspruch aufgelöst. Donald gibt jede Zahl zurück und behält doch immer eine, denn nach jeder folgen noch unendlich viele.

Donald Duck will never become as rich as his uncle Scrooge McDuck. If he gets some money, he soon spends it, except one dollar as an emergency ration.

He starts with two dollars, spends one, gets another one, spends one, and so on. He enumerates his dollar notes with felt pen in order to spend always the oldest dollar. The following sequence shows the stepwise development of the numbers:

1, 2 2 2, 3 3 3, 4 4 ...

Since he, as a cartoon character, lives forever, set theory can be used to calculate the limit of his wealth.

The set theoretical limit shows which numbers Donald will possess forever. It is the empty set.

The set-theoretical interpretation of this limit says that Donald will spend all natural numbers. So the set of numbers which he never spends is empty. (This interpretation is required for enumerating all rational numbers or all lines of a Cantor-list. The set of not enumerated rationals or lines must be empty.)

Fact is however, that Donald always keeps a number. The set of not spent numbers is not empty. Even the limit cannot be less than 1 $.

The solution of this obvious contradiction is this: Set theory proves for every natural number n that the set of not spent numbers up to n is empty. But it is also true that every number n belongs to a finite initial segment upon which infinitely many numbers will follow. This resolves the contradiction: Donald returns every number and nevertheless always keeps one, because every number is followed by infinitely many.

§ 516 On Ducks and Bathtubs

Ben Bacarisse discusses my well known example Scrooge McDuck every day gets enumerated $2 and returns enumerated $1. If he happens to return the right numbers, he will get unmeasurably rich. If he happens to return the wrong numbers, he will go bancrupt. and adds: But Prof. Mueckenheim explicitly tells us that we should not use the limit of the cardinalities as the measure of long-term wealth. http://bsb.me.uk/dd-wealth.pdf

This is not quite true. Of course we can use instead of the cardinality 0 of the empty limit set the limit ¡0 of the cardinalities to measure wealth. But we cannot talk about wealth unless McDuck owns enumerated dollar-notes. If McDuck always returns the smallest nunmber the set theoretic limit is the empty set. This limit only says that of every finite initial segment {1, ..., n} no element forever remains in McDucks hands. But we cannot assume that a property owned by every finite segment {1, ..., n} is automatically inherited by the complete set of natural numbers. If this were true, then McDuck would be rich in the limit without having any dollar. I agree with Ben's conclusion that the wealth of McDuck is measured by the limit ¡0 of the cardinalities, but I do not agree that a person having ¡0 dollars has no dollar. Neither a bathtub can be consideres full without any water molecule being inside. This story illustrates in a very lucid way that there is a difference between a statement "for every number of any finite initial segment" and "for all numbers of Ù". The limit of the sequence of all finite initial segments differs considerably from a complete infinite set Ù. If you agree that wealth without dollars is impossible, then you agree too that the enumeration of all rational numbers never covers all rational numbers. Neither all natural numbers can be used to enumerate a set. If you claim that wealth without dollars and wetness without water is possible, then you push yourself out of any reasonable position - and certainly cannot expect to successfully pass a university of applied sciences. But I do not believe that anybody would claim that.

§ 517 Die Bedeutung des Mengenlimes / The meaning of set limits

Die Folge (sn) der Mengen sn = {n+1, n+2, ..., 2n} {2} {3, 4} {4, 5, 6} ... besitzt den Grenzwert limnض sn = { } (vgl. § 090). Der Grenzwert limnض |sn| der Kardinalzahlenfolge ist unendlich. Dies bedeutet keinen Widerspruch, denn der Mengenlimes enthält lediglich Elemente, die niemals aus den Mengen der Folge verschwinden. Solche Elemente gibt es nicht. Die Kardinalzahlen dagegen geben die Anzahl der Elemente in den Mengen der Folge an. Der Grenzwert zeigt, dass mehr Elemente hinzugefügt werden als verschwinden und insbesondere niemals eine leere Menge vorkommt. Sollten die Grenzwerte jedoch "im Limes" oder "für ω" realisiert werden, dann ergäbe sich ein Widerspruch, wie sich schon an der Notation zeigt, denn sω = { } and |sω| = ¶ erlauben, nicht zwischen limnض |sn| und |limnض sn| zu unterscheiden. Wie diese Realisierung der Grenzmenge zustande gekommen ist, wäre nicht aus ihr selbst erschließbar.

Deshalb können wir feststellen, dass die Vollendung des Unendlichen nicht gelingen kann. Allerdings ist sie für die Mengenlehre unerlässlich. Zum Beispiel beruht Cantors "Beweis" der Existenz transzendenter Zahlen auf einer vollständigen Liste aller algebraischen Zahlen, ebenso wie der "Beweis" überabzählbarer Mengen oder die Ordinalzahl ω + 1 die Vollständigkeit "einfach unendlicher", abzählbarer Mengen voraussetzt.

Consider the sequence (sn) of sets sn = {n+1, n+2, ..., 2n} {2} {3, 4} {4, 5, 6} ... The set limit limnض sn is empty (cp. § 090). The limit of cardinalities limnض |sn| is infinite. This is not a contradiction. The set limit indicates those elements which remain in the sequence forever. There is none. The cardinalities indicate the number of elements in the sets. The limit shows that by an infinite supply forever elements are inserted into the sets such that they never run out of elements. If however these limits should become realized by a set "in the limit" or "at ω", then sω = { } and |sω| = ¶ would show a contradiction as can be seen by the notation already, because for realized limits there would be no distinction between limnض |sn| and |limnض sn|.

Conclusion: Infinity is never finished. But set theory needs this completion, for instance in the "proof" of existence of transcendental numbers by diagonalization of a complete list of all algebraic numbers or for "proving" the existence of sets that are larger than "simply infinite", countable sets or by considering ω + 1.

§ 518 What did Fraenkel wish to express with his story of Tristram Shandy?

The appearance that a set, so to speak, can "contain equally many elements" as a proper subset is in a certain contrast {{that is, so to speak, an understatement}} with the well-known theorem: The whole is always larger than a part of it. This apparent contrast, already clearly recognized by Galilei, has historically been an essential obstacle to the admission of the notion of actual infinity, because it seemed to discredit the infinite sets possessing such a paradoxical property. In reality, however, this theorem of the whole and its part had been proven only in the domain of the finite, and there was no reason to expect, that it would maintain its validity in the giant step that leads from the finite to the infinite {{let alone any reason to accept the contrary}}. Footnote: Even more paradoxical appears the equivalence between two infinite sets of apparently very different perimeter, if it is seemingly transferred into the practical life. The uncomfortable feeling occuring in this case disappears if one realizes that this reality is only ostensible and that our perception is not adjusted to it. Well-known is so the story of Tristram Shandy ... {{cp. §. The uncomfortable feeling does not at all disappear when we realize that the natural numbers have the same well-order as the days or years of Tristram Shandy and that when enumerating the rational numbers always one settled task implies an infinity of further tasks. Always infinitely many natural and rational numbers remain unpaired and there is not the least proof of equinumerousity.}} [Adolf A. Fraenkel: "Einleitung in die Mengenlehre" 3. Aufl., Springer, Berlin (1928) p. 24]

§ 519 Real analysis has its merits in all domains of physics like classical and quantum mechanics, thermodynamics, electrodynamics, special and general theory of relativity, atomic and nuclear physics, astronomy and cosmology. It is further applied in chemistry, biology, medicine, engineering sciences and even many branches of economic. There are only two realms where its application is not beneficial, namely the different branches of theology and transfinite set theory.

§ 520 Set theory as a perpendicular expression of a horizontal desire

The matrix

0.1 0.11 0.111 ... fails to include its limit 1/9 = 0.111... like every strictly monotonically increasing sequence. According to set theory, this matrix contains all ¡0 horizontal rows with at least one 1. But there are not ¡0 such vertical rows. That means set theory gives different answers to this completely symmetrical question. It is inconsistent. Or does it depend on the direction of gravitation? In order to be consistent, there should be as many horizontal as vertical rows. In fact, this is true, since both are potentially infinite ¶. Set theory would only be consistent with ¡0 in both cases. Alas, this is only a wish. So set theory is also "a perpendicular expression of a horizontal desire" (G. B. Shaw).

§ 521 Remarkable sequences of sets and their different limits

(an) with an = {n} has limnض {an} = { }, |limnض {an}| = 1, {limnض n} = ¶ (bn) with bn = {-1/n, 1/n} has limnض {bn} = { }, |limnض {bn}| = 2, {limnض (-1/n), limnض (1/n)} = 0 n n (cn) with cn = {n } has limnض {cn} = { }, |limnض {cn}| = 1, {limnض (n )} = ¶ (dn) with dn = {n/n} has limnض {dn} = {1}, |limnض {dn}| = 1, {limnض (n/n)} = 1 (en) with en = {(1+n)/n} has limnض {en} = { }, |limnض {dn}| = 1, {limnض ((1+n)/n)} = 1

The sequences are constructed by always removing the terms with n and introducing the terms with n+1. In the first three sequences no term stays forever. Only this is expressed by the empty "limit". Applying actual infinity, however, we could "get ready". Then all natural numbers could get "exhausted". The empty sets then are the sets "at ω" Then the limits of the sequences of cardinalities would also apply to the set "at ω", producing a contradiction. Then also numerator and denominator of (dn) would get "exhausted" like that of sequences (cn) and (en), leaving the 0 n empty limit set "at ω" instead of {1}. And what "finishes" the sequences {1n} or {n } or {0 } "at ω"?

§ 522 "Every" is not "all" It is possible for every n in Ù to enumerate the first n rational numbers q1, q2, q3, ..., qn. Set theorists claim that this proves the possibility of enumerating all rational numbers. It is possible for every n in Ù to order the first n rational numbers q1, q2, q3, ..., qn by size. Set theorists do not claim that this proves the possibility of ordering all rational numbers by size. What is the difference?

§ 523 Can the manner of marking influence the result?

Let (sn) be the sequence of sets sn = {n} with n œ Ù. This sequence has an empty limit set.

Let (tn) be the sequence of sets tn = {I1, I2, I3, ..., In} where we have indexed strokes in order to distinguish them. sn+1 comes out of sn by adding stroke number n+1. (A unary system is the historically first manner of marking natural numbers.) This sequence has not an empty limit set. The number of strokes diverges towards ω, the sets of indices diverge towards |Ù.

§ 524 Are finite cardinal numbers natural numbers?

Cantor has shown how the natural numbers can be defined as cardinal numbers of sets. First Zermelo and later v. Neumann have shown how the natural numbers can be defined as ordinal numbers of sets. Zero, the most unnatural number though, has been raped and mutilated to become a "natural" number, only in order to justify the unsound idea that a finite initial segment of the ordered set Ù has a cardinal number surpassing all its elements and to deceive mathematicians with the lie that this is a natural state and therefore cannot be different in |{1, 2, 3, ... }| = ¡0. The Löwenheim-Skolem argument has been perverted by defining what "the system thinks". The countability of all really real numbers has been dampened by imaginating undefinable "real numbers". The impossibility of well-ordering uncountable sets has been overridden by "proving" that the impossible is possible. But all these desperate attempts to keep set theory free of contrsdictions have been without success. Some set theorists have recognized that contradictions nevertheless are unavoidable and now are claiming that finite cardinal numbers are not natural numbers. So mathematics is completely decoupled and isolated from its asserted "basis". That's the best inconsistency proof of set theory, isn't it?

§ 525 Let (sn) be a sequence of sets sn of rational numbers q such that for n = 1, 2, 3, ... sn+1 = (sn » {q | n < q § n+1}) \ {qn+1} with s1 = {q | 0 < q § 1} \ {q1} and q1 = 1/1, q2 = 1/2, q3 = 2/1, q4 = 3/1... the positive rational numbers indexed by Cauchy- diagonalization of the matrix of positive rational numbers. The set sn contains the rationals of the interval (0, n] which have not got an index k § n. When investigating this case for all natural numbers, we get two limits, one for the sequence of sets and one for the sequence of cardinal numbers: limnض sn = { } is indicating that no rational remains without index. limnض |sn | = ¶ is indicating that the set of rational numbers without natural index has infinitely many elements, not only for every sn but also in the limit. My questions: Why is the first limit considered more reliable than the second one? Has the second limit a mathematical meaning? If so what is it? My answers: limnض sn is meaningless since it is impossible to exhaust an infinite set. There is an infinite supply; this is indicated by limnض |sn | = ¶.

§ 526 Smallest possible super task

The limit of the sequence (sn ) with sn = {n} is the empty set. This means, among others, that there is no natural number n that remains in all terms of the sequence. The ordered character of the natural numbers allows us to understand this sequence as a super task, transferring the set Ù from a reservoir A containing Ù via an intermediate reservoir B to the final reservoir C. Every state of B can be represented by a term of the sequence and vice versa.

However, if we introduce the condition that a number n may leave B not before the number n+1 has been inserted into B then we have the same limit, i.e., the whole set Ù will reach C, although this can be excluded by the definition that B always contains at least one element of Ù.

This contradiction shows that the set limit is not a reasonable notion. It indicates that all natural numbers with no exception are in C while this is clearly false.

Note that the additional condition is not an artificial hurdle because the condition that every natural number n is followed by a natural number n+1 is basic to all natural numbers.

§ 527 Limits

A proper limit is a state that is approached better and better by the terms of a sequence.

limnض n n Is 2 = limnض 2 ?

Neither sequence has a proper limit. These limits are improper and their meaning is only that the sequences increase beyond any given real number. Neither exists as a real number. In calculus we cannot decide what ¶/¶ is. But often the unbounded increase on both sides is accepted as being equal as the improper limit oo. Many write 2ÿ¶ = ¶, for instance. Above equality in this sense is obvious, when we refrain from using exponential notation. Then both sides simply read 2ÿ2ÿ2ÿ... = 2ÿ2ÿ2ÿ... so that there cannot be a difference.

Now let sn = {n, n+1, n+2, ...}. Why is 0 = |limnض sn| ∫ limnض |sn| = ¶ ?

Also in this case we have improper limits only, showing a never ending process. limnض sn = { } expresses the fact that n will not be in sets following upon sn. limnض |sn| = ¶ expresses the fact that infinitely many naturals follow upon every n. It is very simple. No contradiction. No exhaustion. And therefore no proof of complete bijection or countability of infinite sets. But many will refuse to understand this because it is so easy to confuse infinite sets with finite sets and to think that infinite sets could be finished and enumerated too.

§ 528 The sequence of singletons {n/(n+a)} has limit { } for a ∫ 0 but limit {1} for a = 0. This is strange. If we refuse to cancel down, then, similar to the case a ∫ 0, the limit should be empty too, because then the natural numbers should also be exhausted.

§ 529 Contradiction

Set theory is based upon the assumption that every positive rational number q gets a natural index n in a finite step of this sequence:

sn+1 = (sn » {q | n < q § n+1}) \ {qn+1} with s1 = {q | 0 < q § 1} \ {q1} and q1 = 1/1, q2 = 1/2, q3 = 2/1, q4 = 3/1... sn is the set of positive rational numbers less than n which have not got an index less than n. The cardinal numbers |sn| = ¶ show that the set of not enumerated rationals is never empty in a finite step n œ Ù. Other steps are not available for indexing purposes. Contradiction, if "every" is interpreted as "all".

§ 530 Why Hessenberg's proof fails in infinite infinity *)

Hessenberg derives the uncountability of the powerset of Ù from the limit-set H of all natural numbers which are not in their image-sets. H cannot be enumerated by a natural number n. If H is enumerated by n, and if n is not in H, then n belongs to H and must be in H, but then n does not belong to H and so on. [Gerhard Hessenberg; "Grundbegriffe der Mengenlehre", Sonderdruck aus den "Abhandlungen der Fries'schen Schule", I. Band, 4. Heft, Vandenhoeck & Ruprecht, Göttingen (1906) § 24 ] If "all" is replaced by "every" and if we keep in mind that every natural number is succeeded by infinitely many natural numbers (and preceded by only finitely many), we get the following sequential explanation of the "paradox": Every set Hk = {n1, n2, ..., nk} containing all natural numbers up to nk, which are not mapped on image-sets containing them, can be mapped by any number m not yet used in the (always incomplete) mapping. This number m is not in Hk and therefore has to be included as m = nk+1 into the set Hk. Doing so we get the set Hk+1 = Hk » {m}. There remain infinitely many further natural numbers available to be mapped on Hk+1. Choose one of them, say m'. Of course, m' is not in Hk+1 and therefore has to be included as m' = nk+2 into Hk+1, such that Hk+2 = Hk+1 » {m'}. This goes on and on without an end. The mapping is infinite. As long as there is no limit-set H, there cannot be a contradiction obtained from not finding a natural number to be mapped on H. ___ (*) This headline sounds rather strange, but it is required to distinguish infinities since Cantor and his disciples have invented finished infinity.

§ 531 What does § 529 show us? 1) The sets sn of the sequence (sn) tell us that always (for all n œ Ù) infinitely many positive rationals § n remain without index § n. So much is irrefutable. The proof holds for all natural numbers. Nothing can index further rationals. But the sets sn are never empty. 2) If all natural numbers n could be used and all sets sn could be constructed, then the finally remaining rationals without index could be indentified. But that is not possible. This is also irrefutable. Both points taken together show that not all natural numbers n can be used and, therefore, not all sn can be constructed. Therefore it is not a logical problem that always something remains. It is simply the exclusive property of infinity, namely to be never finished. What is the advantage of this idea over set theory with its finished infinity, besides that it is the truth? It gets along without finishing the infinite, without exhausting Ù such that an empty set limnض Ù\{1, 2, 3,..., n} remains. It gets along without undefinable "real" numbers, without the necessity to distinguish between finite positiv cardinal numbers and natural numbers, without inaccessible accessories of matheology and without paradoxes of Löwenheim-Skolem or Banach-Tarski. It gets along without an inexplicable discontinuity of the cardinality functions from Tristram Shandy (cp. § 077, § 200) to McDuck (cp. § 515) or |limnض sn| (cp. § 529) that unavoidably always would strike the not initiated thinker. It gets along without an empty limit of the sequence (sn) of, for all n, infinite sets. Everybody not toughened up in a long study of set theory would ask: "How can the infinite sequence have an empty limit? What is the reason? What causes this vacuum?" I think my answer will be accepted by 99 % of all intelligent thinkers. In fact, I have enjoyed this releasing and satisfying experience for many times.

§ 532 It is easy to demonstrate defective enumerations of the positive rational numbers. The mapping Ù Ø –+ is not a bijection, for instance, when all indices n œ Ù are used to enumerate all rational numbers n/1 in –+. The proof presented in § 529 however proves that a purported enumeration fails. By adapting this proof in an obvious way we see that every purported enumeration of –+ is condemned to fail. It is hard to understand how a method could be accepted in mathematics that, depending on the choice of indexing, can (purportedly) enumerate an infinite set but as easily can fail. No scientific application of rational thinking would allow for such an incredible claim. Astounding delusions often have lead astray sectarians to perverse beliefs and actions even such as mass murder http://en.wikipedia.org/wiki/Charles_Manson or mass suicide http://en.wikipedia.org/wiki/Jonestown But mass self-stultification of thousands of assertedly intelligent mathematicians is certainly unique in the history of the whole universe.

§ 533

Cantor's enumeration of the positive rationals –+ (mentioned in a letter to Lipschitz on 19 Nov. 1883) is ordered by the ascending sum (a+b) of numerator a and denominator b of q = a/b, and in case of equal sum, by ascending numerator a. Since all fractions will repeat themselves infinitely often, repetitions will be dropped. This yields the sequence 1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 5/1 1/6, 2/5, 3/4, 4/3, 5/2, 6/1 ... It is easy to see that at least half of all fractions of this sequence belong to the first unit interval (0, 1].

While every positive rational number q gets a natural index n in a finite step of this sequence there remains always a set sn of positive rational numbers less than n which have not got an index less than n (cp. § 529). sn+1 = (sn » {q | n < q § n+1}) \ {qn+1} with s1 = {q | 0 < q § 1} \ {q1} All sn are infinite |sn| = ¶. But also their geometric measure is increasing beyond every bound. This is shown by the following

Theorem. For every k œ Ù there is n0 œ Ù such that for n ¥ n0: (n-k, n] Õ sn. Proof: Let a/1 be the largest fraction indexed by n. Up to every such n at least half of the natural numbers are mapped on fractions of the first unit interval. a is continuously increasing, i.e., without gaps. Therefore n must be about twice as a, precisely: n-1 ¥ 2(a-1) or n ¥ 2a - 1. Examples: a = 1, n = 1 a = 2, n = 3 a = 3, n = 5 a = 4, n = 10 a = 5, n = 12 a = 6, n = 17 ... Therefore for any given k we can take n = 2k. Then the interval (n -k, n ] Õ s . This is satisfied 0 0 0 n0 for every n ¥ n0 too.

This means, there are arbitrarily large sequences of undefiled unit intervals (containing no rational number with an index n or less) in the sets sn.

Remark: It is easy to find a completely undefiled interval of length 101000100000000000 or every desired multiple in some set sn. Everybody may impartially examine himself whether he is willing to believe that nevertheless all rational numbers can be enumerated.

Remark: Cantor does neither assume nor prove that the whole set Ù is used for his enumeration (in fact it cannot be proved). Cantor's argument is this: Every natural number is used, so no natural is missing. He and most set theorists interpret this without further ado as using Ù.

Remark: Although more than half of all naturals are mapped on fractions of the first unit interval, never (for no n) more than 1 % of all fractions of this interval will become enumerated. In fact it can be proven for ever natural number n, that not the least positive interval (x, y] of rational numbers is ever completely enumerated.

§ 534 Set theorists claim that all rational numbers can be indexed by all natural numbers. In § 533 I have shown not only that every natural number n fails but even that with increasing n the number of unit intervals of rationals without any rational indexed by a natural less than n increases without bound, i.e., infinitely. Since nothing but finite natural numbers are available for indexing, and provably all fail, this task cannot be accomplished. I don't know what goes on in the heads of matheologians. But I know that it is deliberately contradicting the magnificent, powerful, and, for all non-matheological purposes, extremely useful mathematics of the infinite that has been devised by Euler, Gauss, Cauchy, and Weierstrass. Rational arguments to straighten these matheological assertions are not available.

§ 535 Solution of Berry's Paradox

Berry's Paradox, first mentioned in the Principia Mathematica as fifth of seven paradoxes, is credited to Mr. G. G. Berry of the Bodleian Library. It uses the least integer not nameable in fewer than nineteen syllables; in fact, in English it denotes 111,777. But "the least integer not nameable in fewer than nineteen syllables" is itself a name consisting of eighteen syllables; hence the least integer not nameable in fewer than nineteen syllables can be named in eighteen syllables, which is a contradiction

There is no paradox, if the correct specifications are added: The least integer not nameable in fewer than nineteen syllables in the usual one-two-three-language is 111,777. It can be named in the more abstract language applied by Berry by 18 syllables. Languages must not be confused. That does not mean that any language should be excluded from mathematics! Only the reduction to one formal language or the invalid assumption that infinite sequences without finite definition can be subject of mathematical discourse, i.e., mathematics, can raise logical mischaps like uncountability. The list

0 1 00 01 10 11 000 ... with all possible meanings (less than ¡0) of every binary word (there are ¡0) contains everything, i.e. every possible notion and meaning that can exist in mathematics. All these meanings belong to one and the same countable set.

§ 536 The diagonal argument depends on representation

Consider a culture that has not developed decimal or comparable representations of numbers. Irrational numbers are obtained from geometrical problems or algebraic equations only. They are defined by the problems where they appear and abbreviated by finite names - just as in human mathematics. If all rational numbers in an infinite list are represented only by their fractions and all irrational numbers by their finite names, it is impossible to apply Cantor's diagonalization with a resulting "anti-diagonal". Such a culture would not fall into the trap of uncountability. (This is erroneous in human mathematics too, because the infinite decimal representation does never allow to identify an irrational number. Note the name decimal-fractions.)

§ 537 The diagonal argument requires that Cauchy sequences of irrational numbers contain their limit.

Cantor's argument constructs from a list (an) of real numbers another real number, the anti- diagonal d, that is not contained in the list. The argument is based on the completion of the anti- diagonal. But this assumption is wrong. The list contains only all finite initial segments d1; d1, d2; d1, d2, d3; ... of d. d itself is not constructed (and cannot be constructed). In a list of all rational numbers, the anti-diagonal should be an irrational number. But it is not. It is only the infinite sequence of all rational approximations. What differs from a list-number is always merely a rational approximation. All these, however, are already elements of the list, by definition. We arrive at a contradiciton, based on the assumption of a rationals-complete list. As a consequence this assumption, that has no foundation in mathematics, has to be rejected.

§ 538 Sequences and Limits

n As the example (1 + 1/n) sufficiently shows, a Cauchy-sequence has infinitely many (¡0) rational elements. Since all terms of all Cauchy-sequence are rational, they belong to the countable set of rational numbers. The limit, if a non-terminating rational or irrational number, differs from the terms of the sequence. Even in case of simple sequences like 0.999... discussions about their meaning have often lead to controversies. 0.999... is simply an infinite Cauchy-sequence. But by writing 0.999... usually the limit is assumed without saying, so that 0.999... = 1.

The same distinction has to be observed with series: n n n ¶ n Σn œÙ 1/2 < 1 but limnض Σ1 1/2 = Σ1 1/2 = 1 n ¶ Σn œÙ 1/n! < e but limnض Σ1 1/n! = Σ1 1/n! = e n! n n! ¶ n! Σn œÙ 1/10 < L but limnض Σ1 1/10 = Σ1 1/10 = L

Ignorance of these differences has lead to the "9-problem" in Cantor-list. Provision has been made that the anti-diagonal cannot have the form 0.999... However, this provision is not necessary. Cantor's diagonal-argument requires more precision than unwritten limits. Every digit appearing in a Cantor-list belongs to a Cauchy-sequence - not to its limit! The Cauchy sequences 1.000... and 0.999... are quite different. The provision shows, however, that set theorists have been confusing sequences and their limits for about one hundred years. (Cantor himself did not make this provision.)

What about writing limnض before every line of a Cantor list? Or what about writing every line of a Cantor-list twice, the second one always equipped with a limnض? Subject and result of diagonalization are always digits, i.e., rational terms of Cauchy-sequences - whether or not these sequences stand for themselves or are used as names of irrational numbers. In a rationals-complete list, this always raises a contradiction. (Here we have assumed the existence of all terms of a Cauchy-sequence. Of course these can never be written. Therefore they can never serve as names or definitions of numbers. For that sake only the finite formulas constructing the infinite sequences and their limits are available.)

By the way, only the confusion about Cauchy-sequence and their limits has lead to the acceptance of Hessenberg's proof of uncountability of the power set of Ù. Every subset M of Ù can be denoted by a sequence like 0.1110010101... having, behind the decimal point, 1 at an index n œ M and 0 at an index n' – M. Being rational numbers, the set of all these Cauchy- sequences is countable whereas Hessenberg meant to have shown the contrary.

§ 539 I claim that every Cauchy-sequence of rational terms with irrational limit does not contain its limit among its ¡0 terms. This is also true if the Cauchy-sequence is the sequence of partial sums of decimal fractions or digits. So every infinite digit sequence is a rational number. Irrational numbers have no decimal representation. Analogously the infinite digit sequence 0.999..., i.e., the whole sum 9/10n over all natural numbers n œ Ù (not as is usually assumed its limit), is a rational number less than 1. Ben Bacarisse objected: You can't define what the "whole sum" means. Remember, in the specific example you gave, it is a specific rational less than 1. Which one? You don't know. You can't say. You can't define the notation. My reply: All natural numbers are finite, although we cannot define a largest one. I can't say either what a smallest number 10-n would be. Nevertheless, nobody doubts that " n œ Ù:10-n is a rational number. This holds for all ¡0 terms if ¡0 is a sensible notion. Otherwise it holds in n potential infinity for all infinitely many n œ Ù. For ΣnœÙ 9/10 to be a rational number less than 1 we need only the fact that all partial sums are rationals less than 1. We need not define it other n than by ΣnœÙ 9/10 .

§ 540 Irrational numbers have no representation as decimal fractions.

Consider the digit seqence (dn)nœÙ of an irrational number like 1/π. This sequence has no limit. But it can be forced to converge by adding factors 10-n. The sequence of decimal-fractions n (dn/10 )nœÙ is a Cauchy-sequence having ¡0 rational terms. The series of decimal-fractions, here written in the usual decimal notation 0.d1d2d3... the partial sums of which 0.d1, 0.d1d2, 0.d1d2d3, ... are ¡0 rational terms of a Cauchy-sequence not containing its limit 1/π. They are in n bijection with the sequence (dn/10 )nœÙ. The series can also be noted by n ΣnœÙ dn/10 . (A) Does it, by magic spell, contain its limit ¶ n Σ1 dn/10 (B) when notation (A) is chosen? Certainly not. Mathematics does not depend on the choice of notation. But how does the digit sequence (B) differ from the digit-sequence (A)?

Answer: The limit (B) = 1/π does not differ from (A) by any digit. But we know that it is not a rational number like (A). Conclusion: An irrational number is not a decimal sequence or series. It has no decimal expansion, it has no representation by digits, not even by infinitely many.

n § 541 Consider the decimal-fractions an = dn/10 of an irrational number x and the Cauchy- sequence of rational partial sums (sk)kœÙ with sk = a1 + a2 + ... + ak.

If the partial sums are written as s1 + s2 + s3 + ... the limit is not present. If the partial sums are written as (((a1) + a2) + a3) + ... the limit is probably not present. If the partial sums are written as a1 + a2 + a3 + ... the limit is probably present. If the partial sums are written as ΣnœÙ an, then the limit is present: ΣnœÙ an = x.

§ 542 What is a number?

This is one of many possible philosophical positions: Numbers are expressions.

Then different expressions are different numbers. Applying mathematical laws (usually base 10 is chosen) and looking into dictionaries we find out that some of these expressions (like 2 and II and two and zwei and 6/3 and 17 - 15 which are different numbers) can be exchanged without changing the meaning of the text. Some expressions, like 2, are more popular than others, like II or 17 - 15. But it is our choice to use any of these expressions which are numbers.

Another philosophical position says that the set of all expressions which can be exchanged without changing the meaning of the text is a number. Then the different numbers belonging to these different expressions are amalgamated into one single number.

No heaven of matheology is required where numbers could exist independent of any mathematical discourse.

§ 543 Two wrong definitions

A definition is wrong if it conveys or establishes wrong information.

The world laughed about Bill 246 which in 1897 passed in first reading the assembly of the state of Indiana. It defined π = 3.2. This wrong definition was certainly welcomed by merchants who had no pocket calculators and disliked the irksome appendix .14.

The world should laugh about the matheological definition of the irrational limit of a rational sequence (an): n Σn œÙ an = limnض Σ1 an *) This definition is wrong because the series a1 '+ a2 '+ a3 '+ ... with purely natural indices, gives the sequence of the partial sums only, all of which are rational and thus are not the limit. This sequence, although written as an infinite sum, cannot simultaneously be the limit. π is not a sum of fractions - not even an infinite one!

*) This wrong definition is necessary in the interest of set theorists who cannot deal in a consistent way with the actual infinite. The following sequence, written as a triangle, has limit 111.... But the limit is not a term of a strictly monotonic sequence. However, if infinity is actual, then the height of the triangle is actually 111... and larger than every line determining its width. This is an inconsistency, not easy to observe and perhaps impossible to understand for matheologians, but present with no doubt.

1 11 111 ... _____ 111...

In order to make this inconsistency disappear, it must be "defined" that the sequence contains or is its limit.

§ 544 In § 542 we have learned about the possibility of defining a number system such that two different expressions cannot denote the same number. I do not know of any obstacle to use this system for common arithmetic and calculus. It is merely a change of names. Instead of talking about different expressions denoting the same number we talk about different numbers which however can be replaced by each other according to the common rules. This may be expressed by the identity symbol too: If x can be replaced by y and vice versa, without changing the indicated value, then we write x = y, otherwise x ∫ y. For example: 1 + 1 = 2 and 1 + 1 ∫ 1.

No changes of mathematics are to be expected - with one exception: The set of all numbers appearing in this system is obviously not uncountable.

§ 545 A matter of notation?

The well-known sequence 0.1 0.11 0.111 ... has been written vertically here, because space on the screen is cheap. It contains all digits and all partial sums of its limit 1/9 - but not the limit itself. When written, on expensive paper, into one line as 0.1, 0.11, 0.111, ... or, on luxurious vellum, even shorter as 0.111..., it is including, as before, all digits that are possible. But in the luxurious version we get the limit in addition.

Simply using another notation includes the limit? Or is it the substratum?? What kind of mathematics could produce such a result???

§ 546 The sequence (Sn) of singletons Sn = {n} has limit limnض Sn = { } with |limnض Sn| = 0. On the other hand limnض |Sn| = 1. What is the final state after all? Has it cardinality 0 or cardinality 1? Why should we trust in one of these results? Is there a contradiction?

A contradiction would appear in fact if the set limit described a final state Sω, as the cardinality limit does. But the set limit does not. It only shows that no natural number remains in the sequence forever.

For every number n there exists a set Sm such that n has left Sm. "n $m: n – Sm This does not allow to conclude that there is ever an empty set appearing. "n: n – Sm

Therefore the cardinality limit which shows that there remain natural numbers in the sequence forever is not contradicted.

Meaning and importance of the cardinality limit is demonstrated in cases where no set limit exists as for circular sequences like (Tn) where Tn = {n mod 2}. It will never be empty as shown by limnض |Tn| = 1.

§ 547 Let's do something constructive. Let's try to collect all possible ways to define irrational numbers (or irrationalities, as others say).

§ 548 It is widely held that irrational numbers can be represented by infinite digit-sequences. We will show that this is incorrect. A digit sequence is only an abbreviated notation for an infinite sequence of rational partial sums. Irrational numbers are limits, incommensurable with any grid of decimal fractions.

It is obvious that strictly monotonic sequences do not assume their limit. Rarely the terms of the sequence and its limit are confused. But this situation changes dramatically when sequences of partial sums of series are involved. It is customary to identify the infinite sum over all terms of a series and the limit of this series.

The sequence of rational approximations 3.1415... is purely rational although we cannot find a fraction m/n = 3.1415... covering all its terms. This disadvatange is shared by sequences like (1/10n) too. We cannot find a fraction covering its infinitely many terms all of which are rational with no doubt. A periodic decimal fraction has as its limit a rational number. A non-periodic decimal fraction has as its limit an irrational number. Conclusion: Irrational numbers have no decimal expansion, no representation by digits, not even by infinitely many. They are incommensurable with every digit-measure. An irrational number needs a generating formula F in order to calculate every digit of the infinite digit sequence S and the limit. The formula F may be interpreted as the number as well as the limit. The implication F fl S cannot be reversed because without F the sequence S cannot be obained.

The mathematical facts discussed above also apply to all sequences of digits (or bits) appearing in the folklore version of Cantor's diagonal argument. Digit sequences are never representing irrational numbers let alone transcendental numbers. Therefore Cantor's diagonal argument does not concern the cardinality of the set of irrational numbers.

For details see: http://www.hs-augsburg.de/~mueckenh/GU/Sequences%20and%20Limits.pdf

§ 549 A name denotes a real number if, when given this name, the receiver can show the real number to be in trichotomy with every rational number. Of course not every receiver can solve this task when given the name. In fact every reader will fail in some step by practical reasons. Further names change over time. Before Jones and Euler had introduced the name π nobody could know that π is the name of a real number. The meaning of a name depends on the applied decoding or language. All this makes the notion of name of a real number informal. Based on this disadvantages the notion "named real number" cannot be used in formal theory. The set of all finite expressions in a given alphabet however can be used in formal theory and can be shown to be countable. In order to obtain an upper estimate for the set of all named numbers, we can assume that there are at most ¡0 languages and that every finite expression is a name of a real number. So every name denotes at most ¡0 real numbers. This results in an upper estimate of ¡0 named real numbers.

§ 550 Surjections and bijections

Cantor, using his first diagonal argument, by enumerating all positive fractions m/n, maps Ù to the set –+ of all rational numbers q such that every rational number q is in the image of infinitely many natural numbers. This mapping is a surjection Ù to –+ but it is tacitly assumed that a bijection can be obtained from it because every infinite subset of Ù can be put in bijection with Ù. So Cantor's mapping is called a bijection Ù to –+.

Dedekind, using the notion of height and the fundamental theorem of algebra, by enumerating all roots r of algebraic equations, maps Ù to the set A of all algebraic numbers x such that every x is in the image of infinitely many natural numbers. This mapping is a surjection Ù to A but it is tacitly assumed that a bijection can be obtained from it because every infinite subset of Ù can be put in bijection with Ù. So Dedekind's mapping is called a bijection Ù to A.

My list of everything, by enumerating all finite expressions u, maps Ù to the set O of all objects of discourse o such that every object o is in the image of infinitely many natural numbers. This mapping is a surjection Ù to O including a surjection from a subset of Ù to the subset — of O, but it is tacitly assumed that a bijection can be obtained from it because every infinite subset of Ù can be put in bijection with Ù. So this mapping can be called a bijection Ù to —.

§ 551 A small-inaccessible-cardinals-axiom

SIC-Axiom: There exist 10 prime numbers the sum of which is less than 9.

Note that this axiom does not prove constructibility but only existence. Using it we can prove that we can find 10 prime numbers the sum of which is less than 9. Of course that does not mean that we can find such numbers - it is only provable that we can find them. Perhaps it will even turn out provably impossible to find these prime numbers and their correct sum. But the axiom should be accepted nevertheless, if not for another reason, then at least as an easily understandable analogon to the axiom of choice.

§ 552 The transfinite hierarchy The theorem of well-ordering has been contradicted by the assumption of uncountable sets. (§ 551) The assumption of uncountable sets has been contradicted by the assumption of countable sets. (§ 535) The assumption of countable sets has been contradicted by mathematics. (§ 533)

§ 553 The elements What is an element? Is it a finite expression? Or is it something supernatural that can be conjured up by a finite expression? In both cases only a countable set of elements can be called up in a given language and, thanks to set theory, in all languages that are suitable to do mathematics, i.e., thinking, talking, and writing mathematics. It is not even necessary to know what mathematics is in order to know this. It is not even necessary to know mathematics in order to understand this.

§ 554 Ein albernes Spiel / A silly game

Die Cantorianer behaupten: Zeige mir eine vollständige Liste aller reellen Zahlen, und ich werde dir eine fehlende reelle Zahl nennen. Gegenbehauptung: Nenne mir eine reelle Zahl, die angeblich in der Liste fehlt, und ich werde sie in die Liste schreiben. Dann folgen Behauptung und Gegenbehauptung mit wachsender Frequenz.

The winner is ... der am längeren Hebel sitzt. Zumindest wenn die Behauptung "in jeder Liste fehlt eine reelle Zahl" mit der Behauptung "ES gibt mehr reelle als natürliche Zahlen" identifiziert wird - wie das die Cantorianer gern tun.

Niemals in der Geistesgeschichte der Menschheit und vermutlich auch niemals sonst hat ein derart simpler Fehler eine so enorme Wirkung hervorgerufen: Die transfinite Mengenlehre, angeblich "die Grundlage aller Mathematik". Doch das messbare Ergebnis ist mager. In Wirklichkeit gibt es keine einzige Anwendung dieser Lehre, weder in der Mathematik, noch in anderen Wissenschaften.

Übrigens, eine Ziffernfolge, also eine Reihe von Dezimalbrüchen bezeichnet niemals eine Irrationalzahl. Zum einen, weil die unendliche Folge nicht komplett genug sein kann (zu jeder Definition gehört ein Endsignal), und zweitens weil n keinen Grenzwert in Ù erreicht.

Eine streng monotone Reihe oder Partialsummenfolge enthält ihren Grenzwert nicht. Beispiel: n k k ΣnœÙ1/10 = (Σ1§k§n1/10 )nœÙ ∫ limnضΣ1§k§n1/10 = 1/9 n denn kein Term der Stammfolge (1/10 )nœÙ ist der Grenzwert. n n " n œ Ù : 1/10 ∫ limnض1/10 = 0

Cantorians claim: Show me a list of real numbers that assertedly is complete, I will find another real number. Counterclaim: Show me a real number that assertedly does not fit into the list. I will put it into the list. Then claim and ounterclaim follow with increasing frequency.

The winner is ... that one who can do the last move. At least if the claim "there is no complete list of real numbers" is identified with the claim "there (somewhere - not exactly to localize) exist more real numbers than natural numbers - as Cantorians like to do.

Never before such a simple mistake has stirred up so much ado. Of course the result is meager. There is no application of this teaching in mathematics or in sciences.

By the way: A digit sequence, i.e., a series of decimal fractions does never denote an irrational number. Firstly, because an infinite sequence cannot be complete enough to determine a number (a definition requires an end-of-file signal), and secondly because n does not reach a limit in Ù.

A strictly monotonic series or sequence of partial sums does not contain its limit. Example: n k k ΣnœÙ1/10 = (Σ1§k§n1/10 )nœÙ ∫ limnضΣ1§k§n1/10 = 1/9 n because no term of the original sequence (1/10 )nœÙ is the limit. n n " n œ Ù : 1/10 ∫ limnض1/10 = 0

§ 555

Es ist offensichtlich unmöglich, die relativen Lagen von 100 Punkten auf der reellen Achse mittels Dreibit-Wörtern zu beschreiben - selbst wenn ein Axiom diese Möglichkeit behaupten würde. Es ist offensichtlich unmöglich, die verschiedenen Lagen von überabzählbar vielen reellen Zahlen in einer Wohlordnung durch abzählbar viele endliche Wörter zu beschreiben - selbst wenn ein Axiom diese Möglichkeit behaupten würde. Trotzdem behaupten die meisten Matheologen diese Möglichkeit mit dem Argument, dass weder im Auswahlaxiom, noch im Wohlordnungssatz von definierbaren oder beschreibbaren Elementen die Rede ist. Das Fehlen dieses Wortes ist nicht verwunderlich, denn eine unbeschreibbare Wohlordnung wäre nutzlos. Cantor führte die Wohlordnung ein, um Mengen Element für Element zu vergleichen, was im Falle unbeschreibbarer und damit unbeschriebener "Wohlordnungen" unmöglich wäre. Zermelo erfand das Auswahlaxiom wonach jede Untermenge einer Menge ausgewählt werden kann. Eine Untermenge auszuwählen bedeutet, sie von allen anderen zu unterscheiden. Dazu muss jede Untermenge beschrieben sein, also eine Beschreibung besitzen. Das ist immer ein endlicher Ausdruck. Aber wie in jeder Religion, folgen die Jünger nicht immer den Lehren der Gründer. Oft werden die Lehren nicht nur verändert, sondern sogar in ihr Gegenteil verkehrt. Im Falle der Matheologie sehen wir nur ein besonders krasses Beispiel der Umkehrung. Es führt jedoch zu einem seltsamen Ergebnis. Da die Elemente überabzählbarer Mengen nur im Denken existieren können, kann auch jede Wohlordnung einer solchen Menge nur dort und nirgendwoanders existieren. Wenn sie im Denken nicht vorhanden ist, so kann sie auch nirgendwo anders gefunden werden. Dies ist für überabzählbare Mengen bisher stets der Fall gewesen. Trotzdem behauptet der Matheologe, in seinem Denken existiere eine Wohlordnung, die nachweislich in seinem Denken nicht existiert. Und da ist noch ein anderes seltsames Ergebnis: Wenn undefinierbare reelle Koordinaten akzeptiert werden, dann enthält jede Menge von 100 Punkten auch undefinierbare Punkte - viel mehr als 99 im Durchschnitt. Und die Standardabweichung ist so klein, dass es immer möglich ist alle Lagen der beschreibbaren Punkte auf der reellen Achse mit Dreibit-Wörtern zu beschreiben.

It is obviously impossible to note the relative positions of 100 points on the real line by three-bit strings - even if an axiom claims it possible. It is obviously impossible to note the different positions of uncountably many real numbers in a well-ordering by a countable set of position descriptions - even if an axiom claims it possible. But it appears possible to most matheologians. Their argument is this: Nowhere, neither in the axiom of choice nor in the well-ordering theorem we find the word "definable". The lack of this word is not astonishing because well-ordering would be useless if it were undefinable. Cantor invented well-ordering in order to compare sets, element by element. This would be impossible for undefinable, hence undefined "well"-orderings. Zermelo invented the axiom of choice, stating that every subset of a set can be chosen. To choose a subset means to distinguish this subset from all other subsets. This implies that every subset has a definition. A definition is a finite expression. But like in every religion, the followers of the originator do not always follow his teachings. They often change it, sometimes even inverting it into the contrary. In case of matheology we see an extremely blatant example of an inversion. However it leads to a strange result: Since the elements of uncountable sets can exist only in the mind, also any well-ordering of such a set can exist only there and nowhere else. If this mind does not know it, the order cannot be found anywhere. This has been always the case for uncountable sets. Nevertheless the matheologian claims to know that he has in his mind a property that he knows he cannot know. And there is another strange result: If the undefinability of real coordinates is acceptable, then every set of 100 points contains undefinable points - far more than 99 in the average. And the standard deviation is so small that it is always possible to note the positions of the definable points on the real line by three-bit strings.