Why Proof by Contradiction Is Unreliable and There Is No Such Thing As an Infinite Set

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Why Proof by Contradiction is Unreliable and there is no such thing as an Infinite Set

Pravin K. Johri

Proof by contradiction, relying on the Law of Excluded Middle, is used universally in mathematics even though it is considered unreliable in other sciences. The most notable application is in the Cantor Diagonal Argument to infer that the unit interval is uncountable. Cantor’s theorem uses it to conclude that there are infinitely many cardinalities of infinite sets. The very idea of “Infinite infinities” should boggle everyone’s mind.

Excerpts from the Wikipedia page “Georg Cantor”

Before Cantor, there were only finite sets (which are easy to understand) and "the infinite" (which was considered a topic for philosophical, rather than mathematical, discussion). By proving that there are (infinitely) many possible sizes for infinite sets, Cantor established that set theory was not trivial, and it needed to be studied. Set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects (for example, numbers and functions) from all the traditional areas of mathematics (such as algebra, analysis and topology) in a single theory, and provides a standard set of axioms to prove or disprove them.

Proof by contradiction was severely criticized by Kronecker, who disliked much of Cantor's set theory because it asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose members did indeed satisfy those properties.

If proof by contradiction is indeed unreliable, it would negate Cantor’s infinite set theory and invalidate many areas in mathematics. This profound conclusion requires background on the genesis of such a method, and this is provided in Appendices A and B.

The Law of Excluded Middle (LoEM)

In logic, the Law of Excluded Middle (or the principle of excluded middle) is the third of the three classic laws of thought. It states that

For any proposition, either the proposition is true or its negation is true.

Excerpts from the Wikipedia page “Brouwer-Hilbert controversy”

The earliest known formulation is Aristotle’s principle of non-contradiction - that of two contradictory propositions (i.e. where one proposition is the negation of the other) one must be true, and the other false. It just seems “logical” that an object of discourse either has a stated property (e.g. “This truck is yellow”) or it does not have that property (“This truck is not yellow”) but not both simultaneously.

The LoEM implies two absolutes, the proposition and its negation, and no gray area or middle (ground) between the two. The likelihood of a middle is excluded in this line of reasoning. Even Quinn [6] acknowledges that the LoEM may not be justified.

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. Instead of debating whether or not it is true, we should investigate the constraints it imposes on our subject.

Given the doubts as to the efficacy of excluded-middle arguments, should they be used in a pure science?

Perhaps there is no clear answer. If that is the case, it is unlikely one can come up with meaningful constraints when such logic applies and when it doesn’t.

Can one analyze whether or not the LoEM is true? The best way to show it is untrustworthy is to demonstrate that it leads to different conclusions than the ones adopted in mathematics.

Complement of a Set

It turns out finite set theory has a basic concept which is quite similar.

In a Venn diagram, a rectangle represents the Universe U universe U which contains all the entities under consideration. Sets are represented by circles and can only be subsets of U. The complement Sc of a set S is S given by the part of the rectangle outside of S's circle.

If an object in not in S it must be in its complement, and vice versa.

Hence, showing x  S is sufficient to establish that x  Sc, and vice versa. It does rely, however, on an implicit assumption that x was in the universe to begin with.

In the previous example, the universe can be all trucks and the set S can be all yellow trucks. Then Sc should be made up of all trucks that are not yellow. But is it?

Proof by Contradiction

A proposition can be established by contradicting its negation.

For example, if the proposition this truck is not yellow is negated then one can establish that this truck is yellow. The postulate this truck is not yellow seems to be simple, specific and straightforward, and so is its negation. They both assert a single property of an object in a clear concise manner. Is the same true for more complicated premises like the ones in the Cantor Diagonal Argument (CDA) and in Cantor’s theorem?

The CDA negates the proposition that the unit interval is countable, which on the surface is also a succinct description. However, below the surface, there are many axioms, definitions, methods, etc. that have allowed one to reach the point when such a claim can be made. Should any or all of these be included in the proposition? Are any implicit assumptions negated by the proof by contradiction argument along with the main proposition?

Failure of Proof by Contradiction

It turns out that even the simplest of propositions need not be absolutely precise.

 What if the truck is red and yellow?

Now, the previously incompatible statements this truck is yellow and this truck is not yellow are both neither entirely true nor wholly false!

 Proof by contradiction failed in the simplest of examples as soon as a  second dimension of multiple colors is introduced.

Note that the original proposition and its prescribed negation didn’t say anything about single or multiple colors. As stated, it should have applied in both cases.

Potential Fixes?

 Perhaps, the proposition needs to include a condition that the truck is of a single color:

IF the truck is of a single color, THEN either it is yellow or it is not yellow.

But now the result only applies to a part of the Universe (trucks of a single color) and not the entire Universe (all trucks) and it is no longer true in general.

 Perhaps, the original proposition has be enhanced to include a second claim that the truck is of a single color in order to rule out the red and yellow truck:

Proposition The truck is of a single color AND it is yellow

Now the proposition is a conjunction of two assertions, and there is more than one negation of such a compound postulate with multiple claims. Either claim, the truck is of a single color or the truck is yellow, can be negated resulting in three possible negations.

Negation 1 The truck is of a single color and it is not yellow Negation 2 The truck is not of a single color and it is yellow Negation 3 The truck is not of a single color and it is not yellow

Some negations, like negation 2, need not be consistent. If the truck is not of a single color, then just specifying that it is yellow is not a sufficient description. However, negations 1 and 3 are valid.

 Which negation does the proof by contradiction argument establish?

Negation in Logic

A conjunction (AND) of logical propositions p  q is true only if both propositions are true. A disjunction (OR) of logical propositions p  q is true if at least one proposition is true. Negation is an operation that takes a logical proposition p to another proposition "not p", written ¬p, which is true when p is false and false when p is true.

The negation of a conjunction of two propositions p and q is defined as ¬(p  q)  ¬p  ¬q The right hand side evaluates to true if: (1) ¬p is true and ¬q is false; or (2) ¬p is false and ¬q is true; or (3) ¬p is true and ¬q is true.

 In general, a proposition with n simultaneous claims has 2n – 1 negations.

The Cantor Diagonal Argument (CDA)

How many logical claims are there in more complicated propositions like the one in the CDA? The CDA makes an explicit assumption that the unit interval is countable, creates a contradiction, and concludes that the unit interval must, therefore, be uncountable. It is normal to expect the explicit assumption caused the violation. However, some mathematicians have asserted that this need not be the case - that the CDA is a contradiction of the concept of actual infinity itself and is denying an implicit assumption instead of the explicit supposition. The CDA is based on many prior assumptions: the concept of actual infinity, the axiom of infinity, the notion of one-to-one correspondence between infinite sets, the Field axioms, etc. The contradiction could easily be indicating that one or more of these concepts is flawed.

A proof-by-contradiction argument, if used improperly, can arrive at a wrong conclusion.

In Appendix C the CDA itself is used to show that the CDA is ill-defined and does not actually examine numbers to infinite digits. Lots of underlying assumptions are, in fact, wrong!

The Two Notions of Infinity and the Axiom of Infinity

Aristotle’s abstract notion of a potential infinity is “something” without a bound and larger than any known number. Excerpts 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). Excerpts 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 at least one infinite set namely the set N = {1, 2, 3 …}.

An actually infinite set has been formed from a potentially infinite sequence 1, 2, 3 … even though these two concepts were contrasted to be different. Completion of the set occurred without a single modification to the incomplete sequence, unlike the completion of a metric space which requires adding the limits to the Cauchy sequences.

Misapplication of Proof by Contradiction in Infinite Set Theory

A set is a fixed collection of distinct objects. Is the object N = {1, 2, 3 …} of an actually infinite size ℵ0? Yes, according to the axiom of infinity. The reasoning is 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

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

The correct conclusion is that N does not have a fixed size and, hence, cannot be a set. It agrees with the fact that the sequence 1, 2, 3 … is endless and does not have a specified size. The proof by contradiction argument actually negates the axiom of infinity instead of corroborating it.

 There is no such thing as an infinite set!

Why did proof by contradiction fail? Is there a middle in mathematics? It actually turns out there is not just one middle but two, and they are related.

The Middle in Mathematics

Mathematics relies on proof by contradiction to establish a 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 …}. However, the sequence 1, 2, 3 … does not increase to ℵ0. The aleph numbers are larger than all natural numbers. There is an undefined gap or a fuzzy gray area in between. This is an outcome of the axiom of infinity which conflates finite with infinite. Many infinite procedures are defined in mathematics. Some are allowed to jump the gap and some are not. The Cantor set and the nested closed interval theorem are examples of procedures that cross the gap. This is covered in another paper.

There is also an undefined gap between convergent sequences and their limit points when the limit is outside the sequence. For example, the sequence {1/n} has the limit 0.

Convergent sequence Undefined Gap Limit point outside sequence

The Archimedean Property ties the two gaps together but also creates a potential issue.

Given any real number x > 0 there is a finite n such that n > 1/x. Given any natural number n, there is an x > 0 such that x < 1/n.

If numbers in a subset of the unit interval are at least a distance x apart, for any x > 0, then there can be no more than 1/x numbers in that subset. And, it cannot be an infinite set. Can the unit interval be uncountable?

Mathematics refers to an undefined “gap” between rational numbers in an effort to explain why Irrational numbers are uncountable.

Irrational Numbers

Almost all real numbers are irrational and they must be denoted with an infinite non-repetitive decimal representation. However, the decimal formula applies only to fixed, finite decimal digits. A number with finite digits is a rational number. If the decimal digits repeat endlessly, as with the number 1/3 = 0.33333…, it is also a rational number. The only option left is that irrational numbers must have non-repetitive infinite digits. Not a single irrational number can be written in the decimal notation as it is not possible to lay out infinite non-repeating decimal digits. Irrational numbers are denoted with symbols like e, π and 2 leading to the biggest absurdity in mathematics that

 Almost all real numbers cannot be written as numbers!

The integers are well-defined discrete points on the real line. Lots and lots of rational numbers lie in the gaps between every two successive integers. The even more numerous irrational numbers must lie in the “gaps” between the rational numbers.

Excerpts from the Wikipedia page “Dedekind cut”

A Dedekind cut is а method of construction of the real numbers. It is a partition of the rational numbers into two non-empty sets A and B, such that all elements of A are less than all elements of B, and A contains no greatest element. If B has a smallest element among the rationals, the cut corresponds to that rational. Otherwise, that cut defines a unique irrational number which, loosely speaking, fills the "gap" between A and B.

In fact, infinitely many irrational numbers must be in every gap between the rational numbers or the irrational numbers cannot be uncountable and constitute a higher order of infinity. This, of course, violates the theorem on the density of rational numbers which states that there is a rational number between every two real numbers. There are many, many, many more rational numbers than integers but they have the same cardinality. Irrational numbers are hard to find vis-à-vis the rational numbers and yet they have higher cardinality.

 Isn’t this an “upside down” reality?

The Supposed Gap between Rational Numbers

The gap between rational numbers may not even exist.

The sequence {1/n} converges to the limit 0. Given any rational number x, the rational sequence {x + 1/n} converges to the rational limit x outside the sequence as all elements of the sequence are strictly greater than x.

There should be a “gap” between the rational number x and all elements of the rational sequence {x + 1/n} or irrational numbers cannot exist. However, if an irrational number x+ were to exist in this “gap” it would be the limit point and not x leading to a contradiction.

Mathematics rationalizes this incongruity by claiming such a x+ cannot be determined exactly. But neither can the set N be determined exactly with all its members laid out without some notation like “…” that is believed to extend to infinity. One can list only a very small subset of the members of the Cantor set, the rational endpoints of the intervals and a few rational interior points, and yet the uncountable Cantor set is believed to exist.

 A pure science is based on inconsistent application of arbitrary rules!

Are Open Intervals Sets?

There is another version of the second middle in mathematics.

Interior of open intervals Undefined Gap Boundary of the interval

It is an outcome of conflicting definitions. To establish the incongruency, one has to fall back to basic definitions. A set is a very simple concept – a fixed collection of distinct objects. All elements of a set should be listed up front and any element can be selected for verifying

membership in a set or for establishing a mapping or for any other purpose. This is a fundamental precept in finite set theory and should continue to apply to infinite sets. Without it there is no set theory.

R has certain properties because its R is also a totally ordered set members satisfy the Field axioms. which has conflicting properties.

The properties of an ordered set have been ignored.

The maximum and minimum elements can be identified in an ordered finite set. A closed interval of R does have a maximum and a minimum element but an open interval does not.

An open interval is a subset of R. Its elements should be fully determined and one should be able to select any element inside it. Once again, reality conflicts with theory.

If one is to select a “minimal point” x in the open interval (0, 1), then there

exists a number y1  (0, 1), such that 0 < y1 < x.

And, there exists a number y2  (0, 1) such that 0 < y2 < y1, and so on, indefinitely. Is this reasonable?

If x, y1, y2  (0, 1) such that y2 < y1 < x, and all three numbers are precisely determined and “visible”, then it is illogical that one would have initially chosen x as the minimal element.

It’s as if these infinitely many numbers y1, y2 … were not available to be chosen when x was selected.

The theory of mathematics is problematic because the Peano and the Field axioms are incompatible with the notion of a set. The Field axiom guaranteeing a unique multiplicative inverse is the root cause of the second middle. This is covered in another paper. The applications of the LoEM in mathematics may suffer from this incongruity in the Field axioms. One cannot absolutely, definitively answer whether the LoEM is correct or not without first fixing the axioms in mathematics.

As shown in this paper, the LoEM has certainly been applied incorrectly to draw the conclusion that the entity N is of an infinite fixed size. Instead, the correct application reveals that the entity N is really of a non-fixed finite size and cannot be a set.

Conclusions

Infinite set theory relies heavily on proof by contradiction but it is unreliable at best.

How can the axiom of infinity be fundamentally wrong?

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.

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 concepts should be

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

Appendix B: Objections to Proof by Contradiction

Mathematicians from three major schools of thought (constructivism and its two offshoots, intuitionism and finitism) opposed Cantor's theories in this matter but their objections were later ignored. Ultimately Hilbert’s formalism and its use of the LoEM resulted in fundamental changes in how mathematics developed.

Excerpts from the Wikipedia page “Constructivism”

Constructivism asserts that 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.

Excerpts from the Wikipedia page “Finitism”

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.

Excerpts from the Wikipedia page “Intuitionism”

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.

Excerpts from the Wikipedia page “Georg Cantor”

For constructivists such as Kronecker, this rejection of actual infinity stems from fundamental disagreement with the idea that nonconstructive proofs such as Cantor's Diagonalization Argument are sufficient proof that something exists, holding instead that constructive proofs are required. Intuitionists hold that mathematical entities cannot be reduced to logical propositions.

Excerpts from the Wikipedia page “Brouwer-Hilbert controversy”

In a foundational controversy in twentieth-century mathematics, L. E. J. Brouwer, a supporter of intuitionism, opposed David Hilbert, the founder of formalism. At issue in the sometimes bitter disputes was the relation of mathematics to logic, as well as fundamental questions of methodology, such as how quantifiers were to be construed, to what extent, if at all, nonconstructive methods were justified, … The fundamental issue is just how does one choose "the axioms"? Until Hilbert proposed his formalism, the axioms were chosen on an "intuitive" (experiential) basis.

Hilbert's axiomatic system – his formalism – is different. At the outset it declares its axioms. But he doesn't require the selection of these axioms to be based upon either "common sense", a priori knowledge (intuitively derived understanding or awareness, innate knowledge seen as "truth without requiring any proof from experience"), or observational experience (empirical data). Rather, the mathematician in the same manner as the theoretical physicist is free to adopt any (arbitrary, abstract) collection of axioms that they so choose.

Indeed Weyl asserts that Hilbert had "formalized it [classical mathematics], thus transforming it in principle from a system of intuitive results into a game with formulas that proceeds according to fixed rules". So, Weyl asks, what might guide the choice of these rules? "What impels us to take as a basis precisely the particular axiom system developed by Hilbert?” Weyl offers up "consistency is indeed a necessary but not sufficient condition" but he cannot answer more completely except to note that Hilbert's "construction" is "arbitrary and bold". Finally he notes, in italics, that the philosophical result of Hilbert's "construction" will be the following: "If Hilbert's view prevails over intuitionism, as appears to be the case, then … the role of innate feelings and tendencies (intuition) and observational experience (empiricism) in the choice of axioms will be removed except in the global sense" – the "construction" had better work when put to the test: "only the theoretical system as a whole ... can be confronted with experience".

Brouwer, in a paper entitled "The untrustworthiness of the principles of logic", challenged the belief that the rules of the classical logic, which have come down to us essentially from Aristotle (384–322 B.C.) have an absolute validity, independent of the subject matter to which they are applied. Brouwer the intuitionist in particular objected to the use of the Law of Excluded Middle over infinite sets (as Hilbert had indeed used it). … Hilbert's adoption of the notion wholesale was "thoughtless" … "that in intuitive (contentual) mathematics this principle is valid only for finite systems."

The “theoretical system” of infinite set theory based on the arbitrary axiom of infinity does not work “as a whole” and is irreparably flawed. It has endured for more than a century and it is time to bury it once and for all

Appendix C: 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 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.

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

[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)

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