WHY a DEDEKIND CUT DOES NOT PRODUCE IRRATIONAL NUMBERS And, Open Intervals Are Not Sets

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WHY A DEDEKIND CUT DOES NOT PRODUCE IRRATIONAL NUMBERS And, Open Intervals are not Sets

Pravin K. Johri

The theory of mathematics claims that the set of real numbers is uncountable while the set of rational numbers is countable. Almost all real numbers are supposed to be irrational but there are few examples of irrational numbers relative to the rational numbers. The reality does not match the theory.

Real numbers satisfy the field axioms but the simple arithmetic in these axioms can only result in rational numbers. The Dedekind cut is one of the ways mathematics rationalizes the existence of irrational 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.

The countable partitions of the rational numbers cannot result in uncountable irrational numbers. Moreover, a known irrational number, or any real number for that matter, defines a Dedekind cut but it is not possible to go in the other direction and create a Dedekind cut which then produces an unknown irrational number.

Irrational Numbers

There is an endless sequence of finite natural numbers 1, 2, 3 … based on the Peano axiom that if n is a natural number then so is n+1. The real numbers are points on the real line and satisfy the Field axioms.

Field Axioms

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

Real numbers can be rational or irrational. The sets N, Q and R are the infinite sets of natural, rational and real numbers, respectively. The theory says that N and Q are countable while R is uncountable, a higher order of infinity, and almost all real numbers are irrational. Further, real numbers can be represented in the infinite decimal notation and irrational numbers are written to infinite non-repetitive decimal digits. This is self-contradictory because

 The decimal formula applies only to fixed finite decimal digits.

Otherwise, the arithmetic in the formula is no longer valid.

The Cantor Diagonal Argument (CDA) does not examine numbers to infinite digits even though there is an explicit requirement in the CDA that all numbers are written to infinite digits (see Appendix D). This requirement in the CDA is unnecessary and does nothing!

The number 1/3 = 0.333… with repeating digits is considered to be written to infinite digits. Repetition results in an unending potential infinite (see Appendix C). Even if it is assumed it can be carried out to “completion”, whatever that means as it is not defined precisely, and one actually ends up with infinite decimal digits, it so turns out that such numbers are also rational.

All numbers written to finite digits are rational. The irrational numbers do not fit the repetitive “infinite” representation. The only option left is that irrational numbers have actual infinite digits but the actual infinite has a nonfinite or transfinite magnitude. Arithmetic cannot be carried out with non-numbers. Even under the assumption there can be infinite decimal digits, no number can be written to non-repetitive infinite digits and, out of necessity, irrational numbers are denoted with symbols like e, π and 2 leading to the biggest absurdity in mathematics:

 Almost all real numbers cannot be written as numbers!

There are very few examples of the irrational numbers and yet they combine to produce a higher-order, uncountable infinity. According to the theory, the CDA establishes that the unit interval is uncountable (although this is not right - see Appendix D).

There is a second infinity, boundlessly larger than the first.

The uncountability of R would be a profound observation were it not contradicted by another result - the rational numbers are dense in the real numbers!

Theorem 1: (Density of the Rationals): Given any two real numbers a, b with a < b, there exists a rational number q such that a < q < b.

Archimedean property: Given any real number x > 0 there is a natural number n such that n > x-1

Proof: The proof is based on the Archimedean property. Since b – a > 0, there is a n such that n > (b – a)-1 and n a + 1 < n b

Let k be the largest natural number such that k  n a and k + 1 > n a

Combining the inequalities n a < k + 1  n a + 1 < n b And q = (k + 1) / n is a rational number in (a, b)

This is a constructive proof in that q is derived from a and b.

The countable rational numbers are densely distributed in the uncountable real numbers. There is a rational number between every two real numbers, which themselves could be rational or irrational leading to the commonsense inference that

 At least half of the real numbers must be rational.

There is a theorem that the irrational numbers are also dense in the real numbers but this is to be expected as the irrational numbers are supposed to be vastly more numerous.

Theorem 2: (Density of the Irrationals): Given any two real numbers a, b with a < b, there exists an irrational number p such that a < p < b.

Proof: When a and b are both positive: From theorem 1, there exists a rational number q in the interval (a/2, b/2). 2q must be irrational because, if 2q is rational, then 2 is also rational. Hence, p = 2q  (a, b)

This proof relies on a pre-existing irrational number such as 2.

Mathematics does not explain how the irrational and rational numbers lie on the real line in accordance with these two theorems. What is abundantly clear is that on the real line,

 There cannot be an interval with just rational numbers in it, and  There cannot be an interval with just irrational numbers in it.

Which, taken together, yields the commonsense inferences

 The rational numbers must be interspersed with the irrational numbers  as points on the real line, each one following the other.

 There cannot be more irrational numbers than the rational numbers!

Where are all the Irrational Numbers?

Almost all real numbers are supposed to be irrational. They should be pervasive but, after listing a handful like e, π, 2, and perhaps other square roots, it is hard to come up with more. Very few irrational numbers have been identified relative to the rational numbers, which begs the question: where are all the remaining irrational numbers?

The integers are discrete points on the real line. Lots and lots and lots of rational numbers lie in the (relatively large) gaps between every two successive integers. 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.

This is the exact opposite of what should be true.

It is like the “upside down” parallel universe in the TV series “Stranger Things”.

Irrational numbers may not exist (see Appendix E). Real numbers satisfy the Field axioms. Starting with the “initial” real numbers 0 and 1, the simple arithmetic in the Field axioms can only lead to rational numbers! How does the first irrational number get “created”? And how do the remaining come about in a way that they completely dominate all other types of numbers?

Construction Models?

Mathematics claims the following construction models yield the irrational numbers:

▪ Construction from Cauchy sequences ▪ Construction from Dedekind cuts ▪ Stevin’s construction

This is just a theoretical claim of “construction” without a single example. There is no method outlined to calculate the irrational limit of a Cauchy sequence of rational numbers. There is no technique specified to compute a new Dedekind cut and, thereby, discover a previously unknown irrational number. There is no procedure prescribed to generate irrational numbers from Stevin’s infinite decimal representation. The logic seems to be - since irrational numbers exist, these methods yield irrational numbers - in theory even if they don’t in practice.

Dedekind Cuts

Mathematics is founded on the core principle of precise definitions (see Appendix A). Why is there only a “loose” characterization of the “gap”?

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 less than x. Similar logic applies to the sequence {x + 1/n} from the other side of x.

There should be a “gap” between the rational number x and rational numbers less than x. However, if an irrational number x- were to exist in this “gap” it would be the limit point of the sequence {x - 1/n} and not x, resulting in a contradiction. Mathematics rationalizes this incongruity by claiming such an x- cannot be determined exactly. But neither can the “gap” in a Dedekind cut be characterized precisely. Why does one exist and the other doesn’t?

The set A must include all rational numbers less than “something”. When the set B is open it must contain all rational numbers greater than the same “something”. When B is closed it must contain all rational numbers greater than or equal to the same “something”. The former yields an irrational number and the latter a rational number. A partition cannot yield more than one irrational number or it violates theorem 1. Hence, half of the real numbers must be irrational and half rational validating the intuitive inferences made earlier.

 How do uncountable irrational numbers result from countable cuts?

For any known real number x, the partition of the reals into the segments (-, x) and [x, ) specifies the corresponding Dedekind cut x. Nothing profound here. The real line can be partitioned into an open and a closed interval at any given point x in it.

The set A includes the rational numbers in the interval (-, x) and the set B contains the rational numbers in [x, ) if x is rational and (x, ) if x is irrational.

 In this case, it is the irrational number that defines the Dedekind cut.

The Dedekind cut would be a useful technique only if one could go in the other direction – partition the rational numbers into two open sets A, B and then discover a previously unknown irrational number. But this has not been demonstrated. So, this “method of construction” is really saying that

IF the rational numbers can be partitioned into two open sets THEN an irrational number is defined to be the boundary between the sets.

The million dollar question becomes whether the rational numbers can be partitioned into two open sets (or open intervals) without first knowing the irrational boundary in between? The partition must satisfy: (1) All elements of A are less than all elements of B; (2) A has no greatest element; (3) B has no least element. Just satisfying condition 1 does not work. The sequence {x – 1/n} has all terms strictly less than its limit x. However, if {x – 1/n}  A and x  B, then B cannot be an open set or x would be an interior point of B leading to a contradiction. Knowledge of elements already in A reveals nothing about the boundary of A because all points in A are interior to A. Can the boundary of A be chosen as the irrational limit x of a convergent rational sequence inside A if the limit is not included in A? This implies knowing the exact value of the limit x in advance, and there is no known method to determine the irrational limit of a rational sequence.

There does not seem to be any way to partition the rationals into two open sets without (pre)specifying the boundary as a known irrational number.

 The only way to define an open interval is to first delineate its endpoints.  A Dedekind cut cannot ever result in an unknown irrational number!

Are Open Intervals Sets?

Excerpts from the Wikipedia pages “Interval (mathematics)” and “Open set” A (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. An open interval does not include its endpoints. A closed interval is an interval which includes all its limit points. Open sets can be defined as those sets which contain a ball around each of their points (or, equivalently, a set is open if it doesn't contain any of its boundary points).

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, which  is taken to be an open set, 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.

There are conflicts in how R is defined which flow over to intervals.

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.

Any element of a set can be selected for establishing a mapping or for any other purpose. This is a fundamental precept in set theory and without it there is no set theory. If the set is

ordered it should be possible to identify the maximum and minimum elements in it. This is certainly true for finite sets. A closed interval of R does have a maximum and a minimum element but an open interval does not.

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.

 Open intervals do not satisfy the requirements of a set.

It is not a surprising conclusion. Neither do N and R. Infinite set theory uses the same definition of a set as in finite set theory (see Appendix B). A set is defined as a fixed collection of distinct objects. All elements of a set should be listed up front. However, both the Peano and the Field axioms dictate that if numbers exist in the sets N and R 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 …

 Neither the entity N nor the entity R is a fixed collection of objects.  Nor can all elements of either entity be listed.  Neither satisfies the definition of a set.

The root cause is the axiom of the unique multiplicative inverse. It allows all points in an open interval to be interior points because there is no longer a minimum value of a real number. It also permits the mapping x  (1, ) → 1/x  (0, 1) to be one-to-one with the outcome that the infinite interval (1, ) has the same cardinality as the finite interval (0, 1), conflating finite with nonfinite.

Conclusions

There is an inherent conflict in the way real numbers are defined via the Field axioms and in the classification of the collection of numbers R as a set.

The Field axioms are incompatible with the notion of a set.

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

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.”

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 principles should be

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

Appendix B: Definition of a Set

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

Infinite set theory uses the same definition of a set as in finite set theory. However, both the Peano and the Field axioms dictate that if numbers exist in the sets N and R 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 …

 Neither the set N nor the set R is a fixed collection of objects.

Cantor’s original set theory resulted in many paradoxes and is considered naïve.

Excerpt from the Wikipedia page “Russel’s Paradox” According to naive set theory, any definable collection is a set. Let  be the set of all sets that are not members of themselves. If  is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell's paradox.

The set of axioms was strengthened to rule out such paradoxes and to tighten results. This produced the non-naïve Zermelo-Fraenkel (ZFC) set theory which, for any property, explicitly disallows that there is a set of all things satisfying that property. However the set of all “selfish” numbers in the proof of Cantor’s theorem is still taken to exist, yet another example of inconsistent application of logic. The correct rule should be

All elements of a set have to 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. N = {1, 2, 3 …} is not a set as the list inside is endless and neither is R. The contorted logic in Russell’s paradox cannot be fully resolved as well.

Appendix C: 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 in mathematics. 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 and incomplete with completed in assuming the object N = {1, 2, 3 …} is a completed 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.

Appendix D: 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 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.

Appendix E: Do Irrational Numbers Exist?

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 and p > q.

 The proof requires that not only p  N but that p2  N as well.

Are all natural numbers in N candidates for the number p if there must always be a significantly larger p2 in N at the same time?

If not, the proof is inconclusive.

For any selected p, it is always the case that the numbers p+1, p+2, … p2 have been excluded as options.

The finite set Nn = {1, 2, 3 … n} has the first n natural numbers in it. 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?

It defies common sense that a property like this could be true for a fixed collection of objects, that is, a set.

As defined, N = {1, 2, 3 …} is an object of a non-fixed finite size and, therefore, is not a set. Since it is half-open, for any selected p there seems to be a p2. But now propositions cannot be established with set theory.

Instead, proof by induction must be used with the natural numbers always – a matter-of-fact stipulation – because they are based on the inductive Peano axiom (if n then n+1 for every n). This would have avoided the whole slew of wrong one-to-one correspondence results in infinite set theory. The proof above fails the induction test.

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)

[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: Dedekind Cuts and Irrational Numbers Why Open Intervals are not Sets Dedekind cuts and Uncountable Irrational Numbers Partitions of the Rationals into Dedekind cuts Why Dedekind Cuts do not work Dedekind Cuts – just a Flawed Theoretical Concept Why the Irrational Numbers are not Uncountable Why Intervals are not Sets Why Intervals of Real Numbers are not Sets

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