Matheology § 001 A matheologian is a man, or, in rare cases, a woman, who believes in thoughts that nobody can think, except, perhaps, a God, or, in rare cases, a Goddess. § 002 Can the existence of God be proved from mathematics? 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 set of real numbers 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) The real numbers can be well-ordered. 6) If this is true, then there must be a being with higher capacities than any human. QED [I K Rus: "Can the existence of god be proved from mathematics?", philosophy.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 mathematicians. 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 number 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 countable set 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 belief 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 subject of proof. Either you believe or not, but this is only a matter of faith. It would be too simple if a proof of existence or non-existence existed. We should not have any choice. – 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 descriptions 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 definition 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 Set Theory", 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 notion "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 definitions, 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 object, including every real number, 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 thought in universities sufficiently rely on the properties of definable numbers. – ANIXX 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 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 time) are non-things. If Koenigs theorem was correct, 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." Today we know that Cantor was wrong and that an uncountable continuum implies the existence of undefinable numbers.