The Flaw in Mathematics Mistakes made in Infinite more than a Century ago

Pravin K. Johri

Cantor’s set theory in the form of the axiomatic Zermelo-Fraenkel (ZFC) set theory is used in all areas of mathematics as a foundational basis for proving theorems. The object N = {1, 2, 3 …} is taken to be a completed, fixed, infinite set. It is really just an assumption, formulated as the of , and is simply based on faith that completed infinite entities exist.

 A fundamental mistake has already been made!

The elements of N, the natural , are all required to be finite and yet the size of N is taken to be an actual infinite quantity denoted as ℵ0 (aleph- null). Finite logic, which manipulates finite numbers, is used to prove properties of infinite objects without any specifics on how this finite logic extends to the actual infinite. This conflates finite with infinite.

It is not clear what ℵ0 is, other than it is not a , it is not finite, and it cannot be measured. A pure science, which should be concerned with numbers only, is now dabbling in ill-defined non-numbers. No wonder the rest of set theory is full of bizarre results, logic that defies common sense and is applied selectively to prove some results but not others, and many outright contradictions.

1

Natural and Real Numbers

There is an endless sequence of finite natural numbers 1, 2, 3 … based on the Peano that 1 is a , and if n is a natural number then so is n+1 for all n.

The real numbers satisfy the Field axioms.

Field Axioms

0, 1 are real numbers and the following exist: A unique additive inverse -x for any x  0 A unique multiplicative inverse x-1, for any real number x  0

Real numbers x + y and x * y, for every two real numbers x, y.

Along with associative, commutative, and distributive properties.

Both the Peano and the Field axioms dictate that if numbers exist then additional numbers exist as well. Neither the natural numbers nor the real numbers can be fully determined. There are always more numbers, and more, and yet more …

Finite Sets

A finite set is defined as a fixed collection of distinct objects.

The size of a finite set is defined as the count of the objects in the set. A fancy term, cardinality, is used to denote size because, later on, some infinite sets will turn out to be uncountable. It is easy to construct mappings between sets as the elements of a finite set are indexed. Mappings are characterized as injective, surjective, and bijective which are all intuitive concepts. There is a single noteworthy theorem to equate the cardinality of two sets if one-to- one correspondence or a bijection can be established between them. A bijection is not necessary to show that two finite sets have the same size. One can simply count the objects in the two sets. It will be required with infinite sets. The best practical use of finite sets is in Venn diagrams.

2

Infinite Sets

The same definition of a set as a fixed collection of distinct objects continues to apply.

The sets N and R are the infinite sets of natural numbers and of real numbers, respectively.

Wade [9] is a standard undergraduate textbook. It starts by implicitly assuming R is a set. “We shall denote the set of real numbers by R.”

Rudin [8] is a standard graduate textbook. It also assumes without reservation that R is a set. “A field is a set S with two operations, called addition and multiplication, which satisfy the so-called field axioms.” “There exists an ordered field R … The members of R are called real numbers.”

The two Notions of Infinity

Aristotle’s abstract notion of a potential infinity is something without a bound and larger than any known number. The sequence of finite natural numbers 1, 2, 3 … is potentially infinite.

A different concept of a completed actual infinity is used with infinite sets. It requires a new abstract notation – the “aleph” numbers to represent the sizes of infinite sets

Excerpt from the Wikipedia page “Brouwer-Hilbert controversy” Cantor extended the intuitive notion of "the infinite" – one foot placed after the other in a never-ending march toward the horizon – to the notion of "a completed infinite" – the arrival "all the way, way out there" in one fell swoop, and he

symbolized this [as] … ℵ0 (aleph-null). Excerpt from the Wikipedia page “Actual infinity” In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the is included) of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given, actual, completed objects. This is contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces a sequence with no last element, and each individual result is finite and is achieved in a finite number of steps.

3

 Why must one “accept” N and R as completed infinite objects?

What exactly is meant by “arrival all the way, way out there"? How can one “arrive” at ℵ0 if the natural numbers do not increase to ℵ0? Why isn’t completion defined precisely as in other areas of mathematics? The completion of a metric space is obtained by adding the limits to the Cauchy sequences. Can completion occur without a single modification to the potentially infinite, incomplete sequence 1, 2, 3 … inside the set N?

Modern Mathematics

Hilbert’s formalism or axiomatic approach is used in modern mathematics and results are established based on the core concepts:

 Precise definitions  Logically correct arguments

According to Quinn this “non-scientific approach” provides “unexpected bonuses”.

Excerpts from Quinn [6] The breakthrough (in mathematics) was development of a system of rules and procedures that really worked, in the sense that, if they are followed very carefully, then arguments without rule violations give completely reliable conclusions. It became possible, for instance, to see that some intuitively outrageous things are nonetheless true.

The definitions include the axioms. An axiom is a fancy word for a starting assumption which is taken to be valid and is not proven. In the past axioms had to be intuitive and also self- evident, but this is no longer required in modern mathematics. Axioms can be arbitrary in Hilbert’s formalism and they have to be stated up front. However, Cantor’s original “naïve” set theory resulted in paradoxes and new axioms were added in the “non-naïve” ZFC to rule out these anomalies, a clear violation of the concept of an axiom!

The existence of infinite sets is cast as an axiom in modern mathematics. This is unfortunate. Axioms are not meant to institute as a fait accompli a proposition which otherwise is hard to establish and may even be wrong.

4

Axiom of Infinity

There exists at least one infinite set namely the set N = {1, 2, 3 …}.

The axiom of infinity is neither intuitive nor is it self-evident. In fact, it contradicts itself.

The set N is considered a completed object even though the sequence of natural numbers inside it is incomplete. The Peano axioms cannot be resolved fully. They result in an endless non-fixed list of numbers which cannot be a fixed collection of distinct objects.

Here is how Fraenkel describes the “actual” infinite set N = {1, 2, 3 …}.

Excerpts from the Wikipedia page “Actual infinity” If the positive number n becomes infinitely great, the expression 1/n goes to naught (or gets infinitely small). In this sense one speaks of the improper or potential infinite. In sharp and clear contrast the set just considered is a readily finished, locked infinite set, fixed in itself, containing infinitely many exactly defined elements (the natural numbers) none more and none less.

The set N is neither readily finished nor is it fixed in itself. There is no sharp and clear contrast, just a glaring self-contradiction. The natural numbers are all finite and yet the set of natural numbers is actually infinite. The natural numbers are called the counting numbers and they steadily increase in steps of 1. If there are infinitely many exactly defined elements then this cannot be true without there being an infinite counting number. But there isn’t one!

 The axiom of infinity conflates finite with nonfinite.

This conflation is the reason behind the many “intuitively outrageous things” or contradictions. It allows finite logic, which doesn’t complete, to be used with infinite objects.

 Just assuming that N is a set and formulating it as the axiom of infinity  does not make it true!

5

Cardinality of Infinite Sets

An infinite set has nonfinite cardinality. It is no longer the count of the objects in the set. The same definition of a set applies to both finite and infinite sets.

 So, why is the size of an infinite set defined differently?

There is a prescription in the concept of actual infinity that “infinity” is represented by an aleph (non)number, a bizarre concept. No one really knows what it means other than an Aleph number is not an ordinary number. It is confusing and wrong to call it a “number”. Ordinary numbers are used to count objects. It is “irrational” that sets such as N and R are made up of ordinary numbers and, yet, their count of elements cannot be such a number and, instead, has to be a non-number! A new way is required to compare the sizes of two infinite sets.

 Infinite sets have fundamentally different properties than finite sets!

They satisfy the “reverse” duck test which says that if it doesn’t look like a duck, doesn’t swim like a duck, and doesn’t quack like a duck, then it probably isn’t a duck.

One-to-One Correspondence

The notion of one-to-one correspondence or a bijection is carried over from finite sets to infinite sets. However, the one-to-one pairing is typically demonstrated only for an arbitrary finite element and not for all elements in the two sets, as in finite set theory. The set of even natural numbers NE = {2, 4, 6 …} is a proper subset of N.

The mapping n  N → 2n  NE is sufficient to show that these two sets have the same (infinite) cardinality.

6

This mapping is an example of finite logic applied to infinite entities. There is an implicit assumption that the natural numbers in the set N are completely enumerated by choosing an arbitrary finite n from it. Is this really true? The selection of a single finite n, no matter how large this n is, still leaves an unending infinite stream of larger natural numbers (n+1, n+2, n+3 …) unaccounted for. And, a finite quantity is insignificant vis-à-vis an infinite quantity. Why doesn’t the Peano axiom (for every n, there is an n+1) figure in some way, shape or form (like induction) in how a bijection is established?

 The cardinality of infinite sets cannot be measured. Nevertheless, one  can conclude two sets have the same cardinality.

How? Based on an arbitrary definition? If the logic is inconsistent it will lead to inconsistent results and, sure enough, it does yield something that is not permissible in finite set theory.

An infinite set can have the same cardinality as its proper subset

Isn’t this a contradiction?

It should be, but it is taken to be true and recharacterized as a counterintuitive result.

Two-to-One Correspondence

In finite set theory, the notion of one-to-one correspondence can be easily generalized to the concept of two-to-one correspondence and even to m-to-n correspondence.

Two-to-one correspondence exists between two sets A and B if every two elements in A are exactly paired with one element in B. As an example, if A = {1, 2, 3, 4} and B = {3, 4}, then the elements 1, 2 in A can be paired with the element 3 in B, and the elements 3, 4 in A can be paired with the element 4 in B. It can be proven that the set A has twice the cardinality as the set B.

7

The mapping n-1, n  N → n  NE (n = 2, 4, 6 …) demonstrates two-to-one correspondence between N and NE and the cardinality of N is twice that of NE. This is a perfectly logical intuitive result. NE only has half the natural numbers as N. In this mapping, the mapped elements in both sets increase at the same rate unlike the mapping n  N → 2n  NE. In “sharp and clear contrast” common sense dictates that the two-to-one mapping is the correct pairing,

The concept of two-to-one correspondence is not accepted in mathematics. There is an arbitrary prescription that only one-to-one correspondence is valid. It is claimed that ordinary numbers and aleph numbers cannot be combined. One cannot talk of “two times the infinite cardinality”. The “same infinite cardinality” really means one times the cardinality. And if infinite cardinalities can be the same or different, what is precluding one from being twice the other?

 The real reason is that it doesn’t fit the theory and is a contradiction!

Induction and One-to-one Correspondence

The natural numbers are defined by the inductive Peano axiom that if n is a natural number then n+1 is also a natural number for all n and, yet, induction is not used to establish one-to- one correspondence.

The one-to-one mapping n  N → 2n  NE is considered valid in mathematics.

 It does not meet the induction criterion!

An inductive proof would show that (1) the mapping is valid for k = 1, and that (2) if the mapping property holds up to some natural number k then it holds for the natural number k+1 as well. Neither the first step nor the second step can be demonstrated for the previous mapping.

For all values of k, N has twice as many (modulo 1) elements as NE.

Mathematicians may justify the mapping by claiming the sets never exhaust. But, then they cannot be completed objects! Mathematicians may also claim that there is no finite natural number in either set that is not mapped to a finite natural number in the other set. Such finite logic is insufficient to prove properties of infinite entities. The axiom of infinity has conflated

8

finite with infinite. The entity N cannot be infinite if all natural numbers inside it are finite! An initial inconsistency just leads to bigger and bigger incongruities as the theory is developed.

NE is a proper subset of N. Intuitively, the mapped elements in the two sets should increase at the same (discrete) rate to infinity or the mapping cannot continue forever. Otherwise one set should exhaust before the other. The mapped elements increase in steps of 1 in the set N and in increments of 2 in the set NE.

There is a mapping that does not have any of these issues.

 With n = 2k, the two-to-one mapping n-1, n  N → n  NE (n = 2, 4, 6 …)  perfectly satisfies the induction criterion.

N has twice as many elements as NE. Any other conclusion defies common sense!

Why the Axiom of Infinity is a Fundamental Flaw

A set is defined as a fixed collection of distinct objects.

All elements of a set should be listed up front. Any rule or condition or notation involved in the formation of a set must be such that it can be fully exercised prior to the formation of the set and all elements of the set determined exactly.

As an example, the entity [1, 2, 3 … 10] can be fully expanded to yield the entire collection of objects [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] which is a valid set. The “…” in the notation decodes to the numbers 4, 5, 6, 7, 8, 9.

The infinite list 1, 2, 3 … is taken to be a complete enumeration of the natural numbers but the “…” no longer translates to the entire specification of all remaining natural numbers. Instead, it now involves the implicit Peano rule (if n then n+1 for all n) and the rule must be applied recursively, without any end in sight.

A change in notation will make the previous point crystal clear.

9

Assume the natural numbers, as defined by the Peano axioms, can be stipulated to be a set. That is,

N  {natural numbers as defined by the Peano axioms} = {1  } The notation  stands for the recursive rule that if n is a natural number then n+1 is also a natural number, for all n.

The symbol  is used instead of the phrase “together with the associated recursive rule” and implies that the rule  is to be applied starting with that number (in this case, the number 1).

This set is denoted as N because the recursive rule  is explicitly incorporated in the definition of the set and not taken to apply tacitly.

All that has been done so far is to enclose the definition of the natural numbers (the Peano axioms) in curly brackets which are used to denote sets.

At this point, the set N has just one element 1   and it is not a natural number. This compound (yet single) element is the natural number 1 with the associated recursive rule .

Let’s try to “readily finish” the set N and “fix” some of the elements, say the first ten natural numbers, just as it is supposed to have happened with the set N according to Fraenkel.

N  {natural numbers as defined by the Peano axioms} = {1  } = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10  }

The recursive rule  was applied starting with the number 1 up to the number 10 to obtain the first ten natural numbers. There are now nine elements which are single natural numbers.

 However, the rule  is still attached to the last number 10.

10

The notation N = {1, 2, 3 …} used in mathematics is equivalent to the notation above.

This is because the extension “3 …” implies that, by virtue of the natural number 3 being present in the set N, we also have the natural number 3+1 present, and the natural number 3+1+1 present, and so on.

This is no different than saying the number 3 has the recursive rule  associated with it.

N = {1, 2, 3 …} is exactly equivalent to N = {1, 2, 3  }

Let’s say the generation of more and more natural numbers was continued from the time Cantor conceived the idea of an actual infinity till today, and it has reached a very, very, very … … … very, very large but still finite number  and the set now looks like

N = {1, 2, 3 … … … … … … … -1,   }

 There is still a funky last element.

The enumerated part of the set 1, 2, 3 … … … … … … … -1 is still just a finite list of natural numbers even if enormously large. The funky last element still represents an infinite list of natural numbers , +1, +2 … and a finite quantity is insignificant vis-à-vis an infinite quantity. Relatively speaking, the enumeration achieved little.

 Can this enumeration be completed so that the rule  disappears?  Completion cannot be based on a belief that it just happens!

[Jules Henri Poincaré] “There is no actual infinity; Cantorians forgot that and fell into contradictions.” [Carl Friedrich Gauss] "I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics.”

The object N is and will always remain a finite set albeit with a funky last element.

11

The assumed set N = {1, 2, 3 …} is sometimes claimed to be infinite as follows:

N is a set Invoking the axiom of infinity Its size cannot be finite The size cannot be n for any n It has an infinite size Invoking proof-by-contradiction

This conclusion contradicts the fact that the endless finite sequence 1, 2, 3 … of counting numbers does not increase to an infinite count.

Here is the correct logic:

N is an object with distinct elements All numbers are different Its size cannot be a fixed finite value The size cannot be a fixed n for any n It is a finite object with a non-fixed size Invoking proof-by-contradiction N is not a set

This conclusion agrees with the fact that the sequence 1, 2, 3 … is endless.

 And, there is no such thing as an infinite set!

Excerpts from the Wikipedia page “”

Finitism … accepts the existence only of finite mathematical objects. The main idea … is not accepting the existence of infinite objects like infinite sets. While all natural numbers are accepted as existing, the set of all natural numbers is not considered to exist as a mathematical object.

This example also demonstrates how proof by contradiction arguments have been misapplied in mathematics.

Excerpts from the Wikipedia page “Constructivism”

For constructivists such as Kronecker … it is necessary to find (or "construct") a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its existence.

12

All kinds of infinite sets, some with rather exotic properties, have been conjured up in mathematics. The Cantor set and the “set of all sets that are not members of themselves” in Russell’s paradox to name a few.

 Sets cannot be based on contorted rules which cannot be fully  resolved to exactly determine all the elements!

So what went so drastically wrong? If the two core concepts

 Precise definitions  Logically correct arguments are “followed very carefully, then arguments without rule violations [should] give completely reliable conclusions”.

A little reflection reveals the deficiency in the prior process. It is not sufficient that definitions (which also include the axioms) are just precise. A new definition must be logically sound vis- à-vis everything that has been developed previously. And, the logic cannot be questionable and arbitrary! It has to conform to common sense. The core principles should be

 Precise definitions which are fully consistent with all prior definitions and results  Logically correct arguments using sound intuitive reasoning

Excerpts from the Wikipedia page “”

The fundamental distinguishing characteristic of intuitionism is its interpretation of what it means for a mathematical statement to be true. In Brouwer's original intuitionism, the truth of a mathematical statement is a subjective claim: a mathematical statement corresponds to a mental construction, and a mathematician can assert the truth of a statement only by verifying the validity of that construction by intuition.

The notion in the axiom of infinity that N = {1, 2, 3 …} is a completed set is inconsistent with the definition of the natural numbers as a finite, never-ending, incomplete list. There cannot be a more glaring discrepancy. Everything built on the arbitrary axiom of infinity inherits this incongruity and is incorrect as well.

13

Irrational Numbers?

Proof by contradiction is used to show that 2 is an irrational number. If 2 is rational, there must exist integers p and q, not both even, such that 2 = p/q. Square both sides to get 2q2 = p2. Since the left hand side is even, p must be even. Therefore, p2 must be divisible by 4 and so q2 must be even as well which contradicts the starting assumption. Hence, no such counterexample exists and 2 must be irrational.

Take the case when p and q are both positive natural numbers and p > q. The proof requires that not only p  N but that p2  N as well. Of course, p2 cannot be equal to the funky last element in N = {1, 2, 3 … … … … … -1,   }.

Are all numbers in N candidates for the number p if there must always be a significantly larger 2 p in N at the same time? Consider the finite set Nn = {1, 2, 3 … n} which is a list of the first n natural numbers. This set can be made as large as one wishes and, intuitively, should approach the object N as n → . The only candidates for the number p in the set Nn are the natural numbers less than n. The fraction of valid candidates, equal to n divided by n, tends to 0 as n → . What happened when the object N became a completed infinite set to reverse this property so that for every p  N there is always a p2  N?

One-to-one Correspondence Arguments?

The mapping n  N → 2n  NE is sufficient to show that these two sets have the same (infinite) cardinality.

This formula is not valid for the funky last element in N = {1, 2, 3 … … … … … -1,   } which represents an infinite list of numbers.

Mathematics assumed that the element “n …” in N = {1, 2, 3 … n …} can be mapped to the element “2n …” in the set NE = {2, 4, 6 … 2n …}. Such a mapping is equivalent to the statement that the unending sequence of numbers n+1, n+2, … can be mapped to the unbounded progression of numbers 2n+2, 2n+4, … and is identical to the original proposition that N is mapped to NE. It is not true unless it is established explicitly.

14

 Finite logic is not sufficient with infinite objects!

The claim that NE, a proper subset of N, has the same cardinality as N is and always will be a contradiction.

 A set cannot have the same cardinality as its proper subset.

There are other fantastic results based on faulty one-to-one correspondence arguments.

Riemann’s Rearrangement Theorem?

If an infinite is conditionally convergent then its terms can be rearranged in a permutation so that the series converges to any given value or even diverges.

The alternating harmonic series is defined as (A) 1/1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7 - 1/8 … = ln 2

The following series is considered a rearrangement of (A): (B) 1/1 - 1/2 - 1/4 + 1/3 - 1/6 - 1/8 + 1/5 - 1/10 - 1/12 … = 0.5 ln 2

The series (B) has half the sum of the series (A) even though it is a rearrangement of (A). An astonishing result if true.

In (B) every positive term is followed by two negative terms which, in turn, are followed by one positive term. It has twice as many negative terms as positive terms.

Are the two series really permutations of each other? Any term (with a finite index n  N) in either series (A) or (B) appears in the other series and provides sufficient justification in mathematics that either series is a permutation of the other. This logic no longer applies if the indices are taken from the set N = {1, 2, 3 … … … … … -1,   }.

15

Riemann’s theorem also violates the commutative property in the Field axioms which states that if x, y are real numbers then x + y = y + x. The commutative property is a rearrangement or a permutation of a pair of numbers and can be applied repeatedly to revert a rearranged series back to its original form.

 A permutation of a series cannot converge to a different sum!

Mathematics justifies Riemann’s theorem by making a fantastic claim – the commutative property does not apply to infinite sums.

 One cannot pick and choose when an axiom is valid and when it is not!

Contradictions cannot be recharacterized as counterintuitive results.

The Cantor Diagonal Argument (CDA)?

The CDA establishes that the real numbers in the unit interval cannot be put into one-to-one correspondence with the natural numbers and the unit interval is uncountable.

 There is a second higher order of infinity.

The CDA uses one-to-one correspondence between the assumed countable enumeration of real numbers and the countable decimal digits to justify the diagonal in its name. This assumption of one-to-one correspondence is flawed and is as big a red flag as there can be. 10n numbers can be written to n decimal digits and there cannot be a diagonal in reality.

The CDA has another issue. It really does not examine any number to infinite digits even though there is an explicit requirement in the CDA that all numbers be written to infinite digits.

This is explained in Appendix A.

16

Cantor’s Theorem?

Cantor’s theorem states that the power set of a set has higher cardinality than the set itself. Given an infinite set, one can recursively form power sets of higher and higher cardinality leading to the conclusion there are infinitely many possible sizes of infinite sets.

When applied to an infinite set such as N, the power set of N becomes uncountably infinite. The proof is by contradiction – assume there is a one-to-one correspondence between N and its power set P(N) and find a contradiction. If one-to-one correspondence has not been defined properly in mathematics, as is indeed the case, then the logic in this proof falls apart. Cantor’s theorem also has another issue which is covered next.

Axiom of Power Set

For every set there exists its power set.

The axiom of power set starts with the existence of just (a single copy of) a set S and this alone is sufficient for its power set P(S) to exist. This contention can be shown to be false.

All elements of a set are distinct objects and there is a single instance of each element in the set – no object is repeated.

A single object can belong to multiple sets. However, whether those sets exist simultaneously (as concurrent objects themselves) depends on what is assumed as the starting state of any problem or theorem - how many copies of the common object are present initially.

Given just a set containing a single copy of each element only disjoint subsets can be formed simultaneously.

Let S = {earth, mars} and its power set P(S) = { {}, {earth}, {mars}, {earth, mars} }. The subsets {earth} and {earth, mars} cannot exist concurrently. That would require two earths a condition not satisfied with just one copy of S. The existence of P(S) requires two instances of the set S! And, in general

17

Theorem: 2n-1 instances of a set with n elements are required to form its power set.

Proof: By induction on n.

Lemma: The power set of a finite set results in no net increase in cardinality.

Proof: By induction on n, the sum of the cardinalities of all subsets in the power set of a set with n elements turns out to be precisely n2n-1. It is the exact product of the cardinality n of the original set and the 2n-1 instances of the original set needed to construct the power set. The count of all objects involved in the formation of the power set remains unchanged and equal to n2n-1. There is no net increase in cardinality.

If a power set with 2n elements can be formed from a set with just n elements it would likely involve magic. On the other hand, if 2n-1 copies of the original set of size n are required then the formation is revealed to be what it really is – a rearrangement. 2n-1 copies of a set of n distinct elements are rearranged to form 2n distinct subsets in the power set of various sizes ranging from 0 to n. There are more (sub)sets in the power set (than the original sets we started with), but they are of smaller sizes, and it is not a profound outcome!

The power set of N can exist only if there are 2 to the power of infinity (minus 1) copies of the infinite set N to start. Such operations with infinite quantities are not allowed in mathematics since infinity is defined as “not a real number”. Cantor’s theorem is meaningless for finite sets |N| and not even defined for infinite sets. The uncountable cardinality 1 = 2 .

 The existence of P(N) presumes there is already an uncountable infinity!

Why was Cantor’s actual infinity needed in mathematics? Did it make some other thorny problems go away and result in a utopia?

[Hilbert] “No one will drive us from the paradise which Cantor created for us”.

It actually turns out that it did. There are even more basic axioms than the axiom of infinity and of power set that are problematic. The Field axiom guaranteeing a unique multiplicative inverse is one culprit. The additive inverse axiom, which results in negative numbers, has issues as well. This is explained in another paper.

18

Conclusions

Real Analysis turns out to be not only counterintuitive but also full of strange concepts, illogical definitions, and lots of contradictions.

Cantor’s infinite set theory is based on lots of flawed assumptions!

It was denounced by many famous mathematicians when it was proposed.

[] “God made the integers, all else is the work of man.” “I don’t know what predominates in Cantor’s theory – philosophy or theology, but I am sure that there is no mathematics there.”

[Jules Henri Poincaré] “There is no actual infinity; Cantorians forgot that and fell into contradictions.”

[Hermann Weyl] “Classical logic was abstracted from the mathematics of finite sets [and applied] without justification, to the mathematics of infinite sets. This is the Fall and original sin of [Cantor's] set theory.”

[L. E. J. Brouwer] [Cantor's set theory is] "A pathological incident in the history of mathematics from which future generations will be horrified."

[Carl Friedrich Gauss] "I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics.”

All of infinite set theory is wrong starting with the concept of an infinite set. No wonder mathematics is so counterintuitive.

The five books listed in the references explain exactly why.

Follow the link “Pravin K Johri papers on Real Analysis” to download more papers by this author.

19

Appendix A: The Cantor Diagonal Argument and Numbers with Infinite Digits

The Cantor Diagonal Argument (CDA) establishes that the unit interval [0, 1] cannot be put into one-to-one correspondence with the set of natural numbers N = {1, 2, 3 …} and, consequently, the set of real numbers R is uncountable.

Suppose there is a complete countable enumeration X = {x1, x2, x3 …} of real numbers in the unit interval [0, 1]. The numbers are written to infinite digits.

x1 = 0.d11d12d13d14……

x2 = 0.d21d22d23d24…… .

Then there exists a number y = 0.d1d2d3d4…… which is in [0, 1] but not in X where

d1 is any digit not equal to d11,

d2 is any digit not equal to d22, . And so on, contradicting the starting assumption.

Can numbers be written to infinite digits? All elements of the infinite sequence 0, 0.3, 0.33, 0.333 … have finite decimal digits. This sequence has as a the 1/3 which is written in the infinite decimal notation as 0.3333… The set S = {0, 0.3, 0.33, 0.333 …} which contains the prior incomplete sequence but not its limit is nevertheless considered an actual, completed, infinite object. When the CDA is applied to the elements of the set S enumerated as this sequence, the diagonal element is always 0 and one choice of y is the limit 0.3333… How did the CDA find a number to infinite digits when the sequence did not have one in it?

The exact value of the non-diagonal decimal digits dij, i  j, is of no consequence to the CDA. Only the diagonal matters in the CDA. So, one can think of a Truncated CDA where all decimal digits dij with j > i are dropped.

CDA Truncated CDA

th Enumeration X = {x1, x2, x3 …} with all Enumeration X’ = {x1’, x2’, x3’ …} with the n numbers written to infinite digits. number in X truncated to n finite digits.

x1 = 0.d11d12d13d14…… x1’ = 0.d11

x2 = 0.d21d22d23d24…… x2’ = 0.d21d22 . .

The diagonal is identical in the two methods and they both find the same numbers y. The requirement in the CDA that all numbers are written to infinite decimal digits is unnecessary! The CDA is ill-defined and wrong. The decimal digit formula applies only to fixed finite digits. There is no such thing as infinite digits and infinite sets.

20

Books Authored by Pravin K. Johri & Alisha A. Johri

[1] Un-Real Analysis: Why Mathematics is Counterintuitive and Impact on Theoretical Physics, Amazon.com, 2016.

Explains why most results in a course on Real Analysis are counterintuitive and often seem contradictory. It examines the axioms in mathematics and identifies the root cause. The concepts of an actual infinity and of an infinite set are flawed. The way one-to-one correspondence is established in mathematics is wrong.

Understanding Modern Mathematics The Power Set & Cantor’s Theorem Complex & Negative Numbers The Cantor Set Infinite Set Theory The Bolzano-Weierstrass Theorem Cardinality of Infinite Sets One-to-one Correspondence in Mathematics Sequences, Series, and Rearrangements of Series Impact on Theoretical Physics Countable and Uncountable Infinite Sets The Root Cause of All Counterintuitive Results Irrational Numbers Summary and Conclusions

[2] The Flaw in Mathematics: Mistakes made in Infinite Set Theory over a Century Ago, Amazon.com, 2016.

Directly develops some of the main results in [1] and includes a more in-depth analysis why some axioms in set theory and Cantor’s theorem are wrong, and how the law of excluded middle has been misapplied.

A Not Uncommon Story Why the Law of Excluded Middle is of No Use Infinite Set Theory The Defect in the Axiom of Infinity Understanding Modern Mathematics The Flaw in the Axiom of Power Set The Numerous Contradictions in Mathematics Conclusions and Impact on theoretical Physics Creating Infinite out of Nothing

[3] Why Mathematics Lacks Rigor: And All of Infinite Set Theory is Wrong, Amazon.com, (2018).

Describes how results are established in mathematics and where this process is deficient. The flaws in Infinite set theory lie in the axioms and in the methodology used to establish rigor. The ensuing contradictory results are simply a consequence of conflicts in the initial specifications. Some logical conclusions are unjustified.

Un-Real Analysis The Error in the Axiom of Power Set Why mathematics lacks rigor Why no one sees the Flaws Why Proof by Contradiction is Ineffective Impact on Theoretical Physics The Mistake in the Axiom of Infinity Conclusions

[4] One-to-One Correspondence between the Irrationals and the Rationals: A Direct Contradiction in Mathematics, Amazon.com, (2018).

Summarizes previous books. Outlines potential issues with Cantor’s infinite set theory which is largely based on arbitrary rules, confounding axioms, and logic that defies intuition and common sense. Establishes one-to-one correspondence between the irrational and the rational numbers in a direct contradiction.

Un-Real Analysis The Counter Argument The Flaw in Mathematics A Direct Contradiction Why Mathematics Lacks Rigor How did Mathematics land up in this Situation? One-to-one Correspondence Leopold Kronecker One-to-one pairing of the Irrational Numbers and the Rational Numbers

21

[5] Why the Cantor Diagonal Argument is Not Valid and there is no such Thing as an Infinite Set, Amazon.com, (2018).

Summarizes previous books. Outlines the various reasons why results in infinite set theory, including the Cantor Diagonal Argument (CDA), are all wrong. The CDA itself is used to establish that the CDA cannot be right. Explains how inconsistent logic is selectively applied in Mathematics to keep the theory whole.

Un-Real Analysis Why the CDA is Not Valid The Flaw in Mathematics There is no such thing as an Infinite Set Why Mathematics Lacks Rigor How Mathematics landed up like this A Direct Contradiction L. E. J. Brouwer The Cantor Diagonal Argument (CDA)

[6] A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today, F. Quinn, Notices of the AMS, 59, No. 1, p. 31-37, PDF, (2012).

[7] Transfinity A Source Book, Wolfgang Mückenheim, (April 2018, previous version March 2017)

[8] W. Rudin, Principles of Mathematical Analysis, Third Edition, (McGraw Hill, New York, 1976).

[9] W. A. Wade, An Introduction to Analysis, Third Edition, (Pearson Prentice Hall, Saddle River, New Jersey, 2004).

Alternate Titles: Why Real Analysis is wrong Why Analysis is wrong Why Real Analysis is flawed Why Analysis is flawed Why Real Analysis is so Counterintuitive Why Analysis is so Counterintuitive The Flaws in Real Analysis The Mistakes in Real Analysis The Flaws in Analysis The Mistakes in Analysis The Flaws in Infinite Set Theory The Mistakes in Infinite Set Theory The Flaws in Cantor’s Set Theory The Mistakes in Cantor’s Set Theory

Keywords: , Infinite Set Theory, Cantor’s Infinite Set Theory, Axiom of Infinity, Actual Infinity, Potential Infinity, One-to-one Correspondence, Countable, Uncountable, Bijection, Rational Numbers, Irrational Numbers.

22