The Cantor Set A Perfect, Uncountable Set with no Segments or A Perfect Example of all that is Wrong with Cantor’s Infinite Set Theory

Pravin K. Johri

The first course in Real Analysis or Analysis is shocking to most students. Mathematics is suddenly counterintuitive and so difficult to comprehend. The course steadily becomes more and more bizarre. Then, towards the end, the student encounters the Cantor set. So far, infinite sets existed and had nonintuitive properties. Now, one can be constructed using an infinite, never- ending procedure of subdividing the unit interval. It involves infinite sequences which previously could not reach their limits but now do, somehow. Mathematics does not specify what changed to allow such a radically different outcome. The subintervals have some characteristics the Cantor set inherits and some it doesn’t without a clear explanation why. There are other unreal properties which are often assigned as homework and one can waste entire weekends trying to make sense of it all.

It is an apt culmination of the course. The Cantor set embodies everything that is wrong with Cantor’s infinite set theory starting from the initial assumptions of an actual infinity and of infinite sets. Both are simply accepted to be true in mathematics, a pure science, and cannot be disproven as there is no proof to start. The only way to show that the theory is totally unfounded is to create one inconsistency after another. And Cantor’s theory is full of contradictions. There is an entire Wikipedia page devoted to it.

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The Cantor Set

Consider subsets of the unit interval [0, 1]:

A1 = [0, 1/3] U [2/3, 1]

A2 = [0, 1/9] U [2/9, 3/9] U [6/9, 7/9] U [8/9, 1] And so on. ∞ Then C = ⋂푛=1 퐴푛 is the ternary Cantor set.

n -n An is made up of 2 intervals each of length 3 and yet the Cantor set contains no segments.

The first step removes a segment of length 1/3, the second a segment of length 2/9, and so on. The total length removed is 1/3 + 2/9 + 4/27 + … = 1. The entire length of the unit interval is excised and yet the Cantor set contains uncountable points.

Excerpts from the Wikipedia page “Cantor Set”

However, a closer look at the process reveals that there must be something left, since removing the "middle third" of each interval involved removing open sets (sets that do not include their endpoints). So removing the line segment (1/3, 2/3) from the original interval [0, 1] leaves behind the points 1/3 and 2/3. Subsequent steps do not remove these (or other) endpoints, since the intervals removed are always internal to the intervals remaining. So the Cantor set is not empty, and in fact contains an uncountably infinite number of points.

It may appear that only the endpoints are left, but that is not the case either. The number 1/4, for example, is in the bottom third, so it is not removed at the first step, and is in the top third of the bottom third, and is in the bottom third of that, and in the top third of that, and so on ad infinitum - alternating between top and bottom thirds. Since it is never in one of the middle thirds, it is never removed, and yet it is also not one of the endpoints of any middle third. The number 3/10 is also in the Cantor set and is not an endpoint. … there are as many points in the Cantor set as there are in [0, 1], and the Cantor set is uncountable … However, the set of endpoints of the removed intervals is countable, so there must be uncountably many numbers in the Cantor set which are not interval endpoints.

It can be shown that there are as many points left behind in this process as there were to begin with, and that therefore, the Cantor set is uncountable.

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Excerpts from the Wikipedia page “Cantor Set”

To see this, we show that there is a function f from the Cantor set C to the closed interval [0, 1] that is surjective (i.e. f maps from C onto [0, 1]) so that the cardinality of C is no less than that of [0, 1].

… For a number not to be excluded at step n, it must have a ternary representation whose nth digit is not 1. For a number to be in the Cantor set, it must not be excluded at any step, it must admit a numeral representation consisting entirely of 0s and 2s. … The function f from C to [0, 1] is defined by taking the numeral that does consist entirely of 0s and 2s, replacing all the 2s by 1s, and interpreting the sequence as a binary representation of a real number.

The Cantor set is also a perfect set in that no point in it is an isolated point.

These properties are truly unbelievable - in that they should not be believed. Ironically, the Cantor set is a perfect illustration of everything that is wrong in Cantor’s infinite set theory.

Inconsistency 1: Is this a valid procedure?

The first issue is whether the Cantor set can be formed in the way it is described.

The sequence of natural numbers 1, 2, 3 … is endless. The process of

forming the subsets An should go on forever. But it terminates and produces a set without a clear description of how the ending comes about.

For segments to disappear the length of the segments, 3-n, must shrink to zero. At the same time, there is no minimum value of a real number and the number 1 cannot be reduced to 0 by repeatedly dividing it by 3.

A clear, precise description is required of what happens when sequences are used within sets to diametrically change their behavior. The way numbers are defined there is an undefined gap or middle (see Appendix D). Some infinite procedures in mathematics are allowed to jump the gap, others are not. No procedure in a pure science should be based on simply the “acceptance” that completion occurs (see Appendix B), and that it can be arbitrarily applied to some situations but not to others.

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Inconsistency 2: The undefined ultimate step (completion)

The Cantor set procedure creates sequences of nested closed intervals.

In = [an, bn], n  N, are nested closed intervals if I1  I2  I3 …

Nested Closed Interval Theorem: ∞ If |In| → 0 as n → , then T = ⋂푛=1 I푛 contains exactly one number.

The sequence of nested intervals is characterized by two sequences {an}

and {bn} with an  bn for all n  N.

Hence, an  a  b  bn and T = [a, b].

Further, T = [a, b] = a = b if |In| → 0.

 This theorem violates the Field axioms!

A positive quantity cannot ever reduce to zero through division.

There is no minimum value of a real number, a direct consequence of the field axiom mandating a unique multiplicative inverse, and 3-n cannot shrink to 0 no matter how large n becomes.

For example, the first interval in each step of the Cantor set procedure has the form [0, 3-n] which becomes progressively smaller as n increases. Each and every one of these nested intervals has positive length and contains uncountable real numbers. Yet, somehow, the ∞ −푛 infinite intersection ⋂푛=1[0, 3 ] of the very same intervals no longer has this property.

If the non-degenerate nested intervals [0, 3-n] shrink to a single point then there must be some, as yet, undefined ultimate step that takes uncountable points (a higher order of infinity than

ℵ0) down to a single point at one fell swoop. This concept is called completion of infinite entities and mathematicians just accept that it happens (see Appendix B) for some entities (intersection of nested intervals) but not others (sequences such as {3-n}). If such an important notion is just based on faith then mathematics is no different than a religion. The onus is on mathematics to precisely define how it happens, in accordance with its core principles (see Appendix A) and remove the inconsistency in how it is applied.

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Inconsistency 3: The use of finite logic

The conclusions regarding the Cantor set rely on finite logic. Many claims are made that are clearly true when the index n is finite.

It is claimed that the endpoints of all remaining intervals in any step of the Cantor set procedure are left behind in the Cantor set because they are not excised in any finite step of the procedure. But these endpoints are all rational numbers and are countable.

An additional claim is made that interior points of intervals are also left behind because interior points such as 1/4 and 3/10 are not removed in a finite number of steps. Almost all members of the Cantor set must be interior points and also irrational numbers, or it cannot be uncountable.

In direct contrast, every step has intervals but none remain at the end.

The sequences characterizing the intervals and their limits need not have consistent properties. The sequence {3-n} has all positive terms but its limit 0 is not positive.

 Finite logic does not necessarily extend to the limiting values!

A convergent infinite sequence like {1/n} can come arbitrarily close to its limit 0 but not quite reach it. However, the infinite intersection of the nested closed intervals defined by the same ∞ sequence and its negative equivalent, ⋂푛=1 [-1/n, 1/n] reaches the single limiting value 0, which is an interior point of all intervals in the procedure. This can only happen if (1) the interval shrinks to a length of zero in some undefined ultimate step and (2) the endpoints of the non- zero interval in the penultimate step somehow disappear.

 Interior points remain only if the endpoints of the intervals disappear!

The way real numbers are defined closed intervals have a peculiar property in that there cannot be a finite number of points ever in a non-degenerate interval.

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A closed interval [a, b] is a single point only if a = b.

If b > a, then [a, b] has uncountable interior points.

Conversely, if there is an interior point in [a, b] then b > a.

 If interior points of intervals remain along with the endpoints of the same  intervals don’t the intervals remain also?

The same finite logic that dictates the endpoints of all intervals and additional interior points remain should also mandate that intervals remain at the end to maintain consistency since there are intervals in every finite step. However, this last inference is rejected in mathematics because it can be shown, via a different line of reasoning, that no segments remain. It so turns out that this divergent conclusion is based on a flawed application of proof by contradiction.

Inconsistency 4: Improper use of proof by contradiction

In Rudin [8], it is shown that no segment of the form [(3k+1)/3m, (3k+2)/3m] remains in the Cantor set. Hence, C contains no segments of a finite size.

Proof by contradiction arguments, such as the one used here, are suspect as there is often more than one valid negation of any proposition. The claim as formulated above is imprecise.

Rudin demonstrates that no interval of a fixed finite size can remain.

And a valid negation is intervals of a non-fixed finite size remain.

The potentially infinite procedure in the Cantor set is never-ending. It is a mistake to assume it completes.

The length of the intervals becomes arbitrarily small without ever reaching a value of zero. The subdivision process cannot end and no conclusion can be drawn as to the final size of the intervals other than that it is not a fixed value.

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Inconsistency 5: Faulty one-to-one mapping

The proof that the Cantor set is uncountable involves a one-to-one mapping from the ternary representation of real numbers to the binary representation. This mapping assumes real numbers can be written to infinite digits. There are actually two differing concepts of infinity (see Appendix B). It is an obfuscation that the same term infinite is used to denote both.

A number written to finite digits is a rational number.

A number written to finite digits with the stipulation that some digits repeat indefinitely, such as 0.333…, is also a rational number.

It is not clear what repeat indefinitely means exactly as the decimal formula only applies to fixed finite digits.

Irrational numbers must have infinite non-repetitive decimal digits (or else they cannot exist).

No irrational number can be written in such a decimal notation and, out of necessity, irrational numbers are denoted with symbols like e, π and 2.

It is the biggest absurdity in mathematics.

 Almost all real numbers cannot be written as numbers!

The Cantor Diagonal Argument (CDA) establishes that the unit interval is uncountable. It requires all numbers to be written to infinite digits. In Appendix E it is shown that this requirement is unnecessary and all numbers in the CDA can be truncated to finite digits with the same outcome. The CDA never really examines a number to “infinite digits”.

Real numbers written to infinite digits is an imaginary concept, based on an unfounded belief that they simply exist. As one cannot write such a number it is impossible to construct explicit examples to either corroborate or to counter any contention. However, conflicts can be easily created with the set of natural numbers N and its subsets (see Appendix D). The finite logic used cannot complete and cover all elements in such sets.

 The way one-to-one correspondence is established is flawed!

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Inconsistency 6: Inconsistent infinite procedures

The Cantor set procedure starts with a single interval. Every successive step doubles the number of intervals to larger and larger finite values (all < ℵ0) in a process that should not end.

However, the process terminates and there are uncountable points (= ℵ1 which is >> ℵ0) left at the end but no intervals. A similar procedure, but in reverse, in the proof of the Bolzano-

Weierstrass theorem (BWT) starts with countable points (= ℵ0) which are repeatedly halved but do not become finite ever nor do they reduce to 1.

BWT: Every bounded sequence has a convergent subsequence.

Start with an interval that contains the entire sequence {xn}. Divide this

interval in two halves. Pick the half-interval that contains elements of {xn} for infinitely many values of n. One of the two half-intervals must satisfy this condition. Select any point from this half interval as the first point of the convergent subsequence. Repeat the process indefinitely to obtain the convergent subsequence.

There are countable points in the sequence and a single interval in the beginning. It is possible the two halved intervals contain an equal number of points (of the remaining sequence) in each step, and so the “number” of points can be thought of as being repeatedly halved. Can the number of remaining points continue to be infinite throughout the procedure?

 Isn’t this a Cantor set procedure in “reverse”?

A finite quantity (number of intervals), starting at the value 1, is repeatedly doubled and becomes uncountably infinite (number of points) eventually in the Cantor set. However, a countable infinite quantity (number of points in the bounded sequence), but infinitely smaller than an uncountable infinite quantity, is repeatedly halved and does not become finite ever in the BWT.

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Conclusions

The Cantor set is claimed to be a perfect, uncountable set with no segments. Instead, it is a perfect example of all that is wrong with Cantor’s infinite set theory

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

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

[Leopold Kronecker] “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.

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Appendix A: Understanding Modern Mathematics

Hilbert’s axiomatic approach is used with arbitrary (neither intuitive nor self-evident) axioms 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.

It turns out that certain logical statements are impossible to contradict and, at the same time, not provable.

Excerpts from Quinn [6] Ironically, … it established “impossible to contradict” as the precise mathematical meaning of “true”.

Proof by contradiction is used to establish a proposition by negating its non-existence. Even Quinn acknowledges such excluded middle logic may be suspect.

Excerpts from Quinn [6] Excluded-middle arguments are unreliable in many areas of knowledge, but absolutely essential in mathematics. Indeed we might define mathematics as the domain in which excluded middle arguments are valid.

It is not sufficient that definitions, which include the axioms, are just precise. A new definition must not conflict with anything that has been developed so far. The logical reasoning must be robust and not based on questionable principles.

The core principles should be

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

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Appendix B: Completion and the Two Concepts 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 axiom of infinity 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.

Axiom of Infinity: There exists infinite sets including the set N = {1, 2, 3 …}.

 A pure science is based on the belief that imaginary infinite entities exist!

The notion of completion is applied inconsistently. A convergent sequence like {1/n} does not complete. It can come arbitrarily close to its limit 0 but not reach it. However, the same sequence inside a set {1, 1/2, 1/3 …} results in a completed set even though the limit point 0 is not in the set. In a third entirely different definition, the completion of a metric space is obtained by adding the limits to the Cauchy sequences.

The axiom of infinity conflates finite with infinite in assuming the object N = {1, 2, 3 …} is a set of fixed infinite size. Its elements are an unending incomplete sequence of finite natural numbers 1, 2, 3 … and it has a non-fixed finite size. This axiom could not be more wrong and is the fundamental flaw in Cantor’s infinite set theory. It is mistakenly believed that finite logic is sufficient with infinite entities and that the choice of an arbitrary n  N is sufficient to enumerate all elements of N. The inconsistencies immediately show up when one-to-one correspondence is established between N and its subsets using this finite logic.

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Appendix C: 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.

Let NE = {2, 4, 6 …}  N be the set of even natural numbers.

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

There is an inherent 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 stream of larger natural numbers (n+1, n+2, n+3 …) unaccounted for. 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.

If the logic is inconsistent it will lead to inconsistent results and, sure enough, it does yield something right away that is not permissible in finite set theory.

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

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

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 easily proven that the set A has twice the cardinality as the set B.

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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.

The concept of two-to-one correspondence is not accepted in mathematics. Why? There is an arbitrary prescription that only one-to-one correspondence is valid.

Induction and One-to-one Correspondence

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

The one-to-one mapping n  N → 2n  NE does not meet the induction criterion! Neither the first nor the second inductive step can be demonstrated for this mapping. For all values of k, N has twice as many (modulo 1) elements as NE. In fact, induction establishes that the cardinality of N should be twice that of NE.

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!

Mathematicians may justify the one-to-one mapping by claiming the sets never exhaust. But, then they cannot be completed objects! And, it cannot be claimed that all elements are mapped!! 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. This is true but such finite logic is insufficient to fully characterize properties of infinite entities. It is taken to be valid only because the axiom of infinity has conflated finite with infinite.

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 and one set should exhaust before the other. In the one-to-one mapping, the mapped elements increase in steps of 1 in the set N and in increments of 2 in the set NE. In direct contrast, they increase at the same rate in the two-to-one mapping.

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Appendix D: The Middle in Mathematics

Mathematics relies on proof by contradiction to establish a logical proposition by negating its non-existence. This method of proof is based on the Law of Excluded Middle which states that either a proposition is true or its negation is true. There is no middle ground between these two absolutes. But there is a middle in the way numbers are defined!

Natural numbers 1, 2, 3 … Undefined Gap Sizes of infinite sets ℵ0, ℵ1, ℵ2 …

ℵ0 is the size of N = {1, 2, 3 …}. The sequence 1, 2, 3 … does not increase to ℵ0. The aleph numbers are larger than all natural numbers and there is an undefined gap between the two. This is an outcome of the axiom of infinity which conflates finite with infinite.

There is also an undefined gap between convergent sequences and their limit points when the limit is outside the sequence.

Convergent sequence Undefined Gap Limit point outside sequence

Some infinite procedures in mathematics are allowed to jump the gap and some are not. Typically finite logic with the index n  N is used to define the finite steps in these procedures and then, somehow, the procedure “completes” and assumes limiting values. The Cantor set and the nested closed interval theorem are examples of such procedures.

This second middle also manifests itself in open intervals and their boundaries.

Interior of open Interval Undefined Gap Boundary of the interval

According to the theory, all points in an open interval such as (0, 1) are interior points. Yet, they are totally ordered and lie on the one-dimensional real line R with the boundary points 0 and 1. There should be points adjacent to the boundaries but there aren’t any.

The concept that all points are interior points in an open interval makes no common sense. Consider a fenced-in section of the beach. The sand is like an open set, literally uncountable particles of sand lie within the boundary of the fence. A sand particle lies in the interior if there are other grains of sand around it.

All particles of sand are not interior to the fence! There clearly is a sand boundary as well – the particles of sand that touch the fence.

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Appendix E: 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 limit the rational number 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 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.

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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 Infinities 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

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[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).

Alternate Titles: Why Proof by Contradiction is Wrong Why Proof by Contradiction is Flawed The Flaw in Proof by Contradiction The Mistake in Proof by Contradiction Why the Law of Excluded Middle is Wrong Why the Law of Excluded Middle is Flawed The Flaw in the Law of Excluded Middle The Mistake in the Law of Excluded Middle The Flaw in the Axiom of Infinity The Mistake in the Axiom of Infinity

Keywords: Georg Cantor, Infinite Set Theory, Cantor’s Infinite Set Theory, Axiom of Infinity, Actual Infinity, Potential Infinity, Law of Excluded Middle, Proof by Contradiction, Negation in Logic, Cantor’s theorem, Cantor Diagonal Argument

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