Un-Real Analysis Why Mathematics Is So Counterintuitive

Un-Real Analysis Why Mathematics is so Counterintuitive

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. In comparison, Calculus was a piece of cake. Now the professor seems to be speaking an entirely different language.

This course starts with the development of finite set theory. A finite set is a very simple concept and easy to comprehend. Then, an infinite set is introduced and it is all downhill from this point onwards.

This paper lays out many of the concepts in Real Analysis. Some are clear and intuitive, some are strange, and some are just bizarre.

Finite Sets

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

The size or cardinality of a finite set is defined as the count of the objects in the set. There is a single noteworthy theorem to equate the cardinality of two sets if a bijection or one-to-one correspondence can be established between them. The best practical use of finite sets is in Venn diagrams.

Infinite Sets and the Two Notions of Infinity

An infinite set is introduced as a set that is not finite. This definition is valid only if the theory developed so far included both finite and infinite sets. But it didn’t! So, what is an infinite set?

 Is it a finite set that is not finite?

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 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.

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

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

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

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

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.

What exactly is meant by “arrival all the way, way out there"? Why isn’t completion defined precisely as in other areas of mathematics (see Appendix A)? The completion of a metric space is obtained by adding the limits to the Cauchy sequences.

The existence of infinite sets is an assumption in mathematics.

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

The set N is a closed completed object but the sequence of natural numbers inside it is incomplete or endless. The natural numbers are all finite and yet the set N is actually infinite.

 The axiom of infinity conflates finite with nonfinite.

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 …

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

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

All elements of a set should be listed up front. Any rule or condition or notation involved in the formation of a set must be such that it can be fully exercised prior to the formation of the set, and all elements of the set determined exactly. As an example, the entity [1, 2, 3 … 10] can be fully expanded to yield the entire collection of objects [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] which is a valid set.

The Peano axioms and the Field axioms cannot be worked out to “completion”. They result in an unending list of numbers and are not a fixed collection of distinct objects. Just assuming that N is a set and formulating it as the axiom of infinity does not make it true!

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

 Can sets be based on contorted logic that cannot be fully resolved?

Cardinality of Infinite Sets

An infinite set has nonfinite cardinality. It is no longer the count of the objects in the set.

 Why is infinite cardinality defined differently?

There was a prescription in the concept of actual infinity that the “size” of an infinite set is represented by the aleph number which is different than an ordinary number. The primary reason for the existence of ordinary numbers is to count objects and to measure quantities of physical entities. It is “irrational” that sets such as N and R are made up of ordinary numbers and, yet, their count cannot be a number!

A new way has to be developed to compare the sizes of two infinite sets.

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 …} be the set of even natural numbers. It is a proper subset of N.

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

There is an implicit assumption that the natural numbers in the set N are fully 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 an inductive argument) in how a bijection is established?

If a method is improperly defined it will lead to inconsistent results and, sure enough, it does yield something that is not permissible in finite set theory.

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

This is a contradiction as clear as it can be. And yet it is taken to be true and recharacterized as a counterintuitive result. The same definition of a set applies to both finite and infinite sets.

 Why do infinite sets have fundamentally different properties?

The cardinality of infinite sets cannot be measured.

 Nevertheless, one can conclude two sets have the same cardinality.

The notions of one-to-one correspondence between infinite sets and the “same cardinality” are arbitrary at best and totally flawed at worst.

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

Two-to-One Correspondence

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

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

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 should be 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 a stipulation that only one-to-one correspondence is valid.

It is claimed there is no such thing as “two times infinity”. It is a made-up arbitrary rule with no basis in reality. Imaginary infinite objects with make-belief “actual” sizes have been “accepted” to exist in a pure science without a single real example of any such object.

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

Induction and One-to-one Correspondence

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

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

 It does not meet the induction criterion!

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

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

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

Mathematicians may justify the mapping by claiming the sets never exhaust. But, then they cannot be completed objects! Mathematicians may also claim that there is no finite natural number in either set that is not mapped to a finite natural number in the other set. Such finite logic is insufficient to prove properties of infinite entities. The axiom of infinity has conflated finite with infinite. The entity N cannot be infinite if all natural numbers inside it are finite! An initial inconsistency just leads to bigger and bigger incongruities as the theory is developed.

There is a mapping that does not have any of these issues. With n = 2k, the two-to-one mapping n-1, n  N → n  NE (n = 2, 4, 6 …) perfectly satisfies the induction criterion.

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

Riemann’s Rearrangement Theorem

The terms of a conditionally convergent series can be rearranged in a permutation so that it converges to any given value or even diverges.

The alternating harmonic series is defined as (A) 1/1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7 - 1/8 … = ln 2 The following series is considered a rearrangement of (A): (B) 1/1 - 1/2 - 1/4 + 1/3 - 1/6 - 1/8 + 1/5 - 1/10 - 1/12 … = 0.5 ln 2

In (B) every positive term is followed by two negative terms which, in turn, are followed by one positive term. It has twice as many negative terms as positive terms. Are the two series really permutations of each other? Any term (with a finite index) in either series (A) or (B) appears in the other series. This supposedly establishes one-to-one correspondence and provides sufficient justification in mathematics that either series is a permutation of the other.

 Riemann’s theorem violates the commutative property.

The commutative property (If x, y are real numbers then x + y = y + x) in the Field axioms involves the rearrangement or permutation of a pair of numbers. This property can be applied repeatedly to revert a rearranged series back to its original form. The terms in a series all have finite indices. The position of any two terms can be switched in a finite number of steps. The original series and the rearranged series must either have the same sum or both must diverge.

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

Mathematics justifies Riemann’s theorem by making an astonishing claim - the commutative property does not apply to infinite sums.

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

Irrational Numbers and Numbers Written to Infinite Decimal Digits

Almost all real numbers are irrational and they must be denoted with an infinite non-repetitive decimal representation.

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

The number 1/3 = 0.333… with repeating digits is considered to be written to infinite digits. Repetition results in an unending potential infinite. Even if it is assumed it can be carried out to “completion”, whatever that means, 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, relatively speaking, very few examples of the irrational numbers and yet the irrational numbers combine to produce a higher-order, uncountable infinity. The Cantor Diagonal Argument (CDA) proves that the unit interval is uncountable.

There is a second infinity boundlessly larger than the first.

|N| The cardinality of the set of real numbers R is taken to be ℵ1 = 2 where |N| = ℵ0 is the infinite cardinality of the set of natural numbers and also that of the set of rational numbers Q. The second higher-order infinity |R| is exponentially larger than the first countable infinity |N| or |Q|. The irrational numbers are infinitely more numerous than the natural numbers or the rational numbers. A finite analogy shows how improbable this conclusion really is.

There is a story related to the invention of chess. The king was so pleased that he asked the inventor to name his prize. The inventor snidely asked the king for one grain of wheat for the first square on the chess board, two for the second, four for the third, etc. doubling the amount for each additional square. The king unaware the amount would grow exponentially agreed to the request but failed to satisfy it. The inventor wound up becoming the new king. The number of grains required = 20 + 21 + … + 263 = 264 – 1 and 264 = 18,446,744,073,709,551,616  18 x1018

By analogy, |N| or |Q| is like an “infinite” chess board while |R| is the number of grains of wheat on this “infinite” chess board.

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.

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.

Continuing with the chess board analogy, think of N as containing just 64 natural numbers. A rational number is expressed as the ratio of two integers and there can be at most 642 = 4,096 positive rational numbers, if numbers like 2/1, 4/2, 8/4 … are taken to be distinct. On the other hand, there should be exponentially more, that is, 264 real numbers. Can ~104 rational numbers be dense inside ~1019 real numbers? The gap becomes exponentially wider as the size of N is increased.

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.

Mathematics does not explain how the irrational and rational numbers lie on the real line in accordance with these 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.

The intuitive implications are

 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 which begs the question: where are all the remaining irrational numbers?

Real numbers satisfy the Field axioms. Starting with the “initial” real numbers 0 and 1,

 The operations 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 that irrational numbers exist and, hence, these methods yield irrational numbers in theory (even if they don’t in reality).

Dedekind Cuts

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. Every real number, rational or not, is equated to one and only one cut of rationals.

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.

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, leading to 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 intervals (-, x) and [x, ) specify the corresponding Dedekind cut x. Nothing profound here. The real line can be partitioned into an open and a closed set at any given point x in it. And, if x is irrational then

 It is the irrational number that defines the Dedekind cut and not the other  way around.

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 without first knowing the irrational boundary in between and then discover a previously unknown irrational number. But this has not been demonstrated.

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 limit point x of a convergent rational sequence inside A if the limit is outside the sequence and is not included in A? This implies knowing that the limit is not in A which is possible only if the boundary of A is already known.

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!

The Root Cause

The root cause lies in a more fundamental concept that is ill-defined.

 An open interval is not a set!

All points in an open interval such as (0, 1) are interior points by definition. Yet, they are fully 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.

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.

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.

Are Open Intervals Sets?

A set is defined as 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.

All elements of R should be precisely determined if it is a set.

An open interval is a subset of R. Its elements should also 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.

Given an x  R, all elements of R which are smaller than x and all elements of R which are larger than x should be exactly ascertainable as well. This applies to every member of R. Hence, if R is indeed a totally ordered set it should be possible to pick the elements in R “adjacent” to x, denoted as x− and x+, where

x− is the largest member of R less than x, and x+ is the smallest member of R larger than x.

The way the real numbers are defined via the Field axioms these “smallest” and “largest” numbers cannot be determined. The existence of x− and x+ also violates the theorems on the density of the rationals and the irrationals.

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

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

Mathematics has to either alter these axioms to make them satisfy the notion of a set or find a different mechanism (other than sets) to study the numbers.

Uncountability and the Archimedean Property

Archimedean Property: Given any real number x > 0 there is a finite natural number n such that n > 1/x.

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.

The CDA asserts that the unit interval contains so many numbers they cannot be mapped one-to-one to the countably infinite set N. A real number can be made arbitrarily small and this, hypothetically, allows uncountable points in the unit interval that are “exponentially more

than countably infinite”. However, the absolute difference between any two distinct real numbers must be positive. But with positive distances the Archimedean property applies.

 Can any subset of the unit interval ever be an uncountable infinite set?

The difference between any two numbers can tend to 0 but it can never equal 0. At every point in every positive monotone sequence tending to 0, the inverse of that point is bounded by a finite natural number. Hence, every such sequence produces a corresponding list of “inverses” which are non-decreasing finite natural numbers. At best, as any sequence tends to zero, the list of inverses can diverge to the first infinity |N|. How do the finite inverses stop being finite, leapfrog the first infinity, and go to the second, exponentially larger, infinity |R| as is claimed in mathematics?

Other issues with Infinite Set Theory

There are more issues with the Field axioms, existence of irrational numbers, proof by contradiction arguments, the CDA, the axiom of Power set and Cantor’s theorem, the Cantor set etc. which are not dealt with here but can be found in references [1-5].

Infinite set theory is built on the flawed foundation of an actual infinity and the self-contradictory concept of an infinite set of finite natural numbers. The teetering pile of ensuing results, aptly represented by “Jenga blocks”, has lots of holes and will topple soon enough.

Conclusions

Real Analysis turns out to be not only counterintuitive but full of strange concepts, illogical definitions, and lots of contradictions. One could call it an “upside down reality” but infinite sets do not really exist and are imaginary.

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 principles should be

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

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: Why Real Analysis is so Counterintuitive Why Analysis is so Counterintuitive Real Analysis The Flaws in Real Analysis The Mistakes in Real Analysis The Flaws in Analysis The Mistakes in Analysis

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