WHY CANTOR’ THEOREM IS MEANINGLESS and there aren’t Infinite

Pravin K. Johri

Cantor’s theorem: The Power of a set has higher than the set itself.

Given an , one can recursively form Power sets of higher and higher cardinality leading to the conclusion there are infinitely many possible sizes of infinite sets. The notion of a single is abstract. The very idea of “Infinite infinities” should boggle everyone’s mind.

Excerpts from the Wikipedia page “” Prior to this (Cantor’s) work, the concept of a set was a rather elementary one that had been used implicitly since the beginnings of , dating back to the ideas of Aristotle. No one had realized that had any nontrivial content. 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 to prove or disprove them.

It turns out that there is a hidden assumption in the of which is equivalent to the theorem’s conclusion. By assuming that Power sets exist via the ,

 Cantor’s theorem presumes the ultimate outcome and is meaningless for  finite sets. Furthermore, it is not even defined for infinite sets!

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The Two Notions 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 “” 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 , and each individual result is finite and is achieved in a finite number of steps.

Few people realize that Cantor’s notion of a “completed infinity” is different than Aristotle’s idea of a potential infinity. It requires an undefined act of “completion” and a new abstract notation – the “aleph” numbers which represent the sizes of infinite sets.

Cantor’s Theorem and Proof

For any set S, the Power set of S denoted as P(S) has a strictly greater cardinality than the set S itself.

When applied to an infinite set such as the of natural numbers N, the Power set of N becomes uncountably infinite.

The result is obvious for finite sets since the size of the Power set of a with n elements is 2n. The proof for the infinite set N is by – assume there is a one-to-one correspondence between N and P(N) and find a contradiction.

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Given any one-to-one correspondence, a in the set N can be paired with a of N in P(N) that either contains or does not contain that natural number. If the former is true, the number is called “selfish” and called “unselfish” otherwise. The set of all unselfish numbers, denoted as D, provides the contradiction.

This set D must be paired with a natural number, say d. By definition, d cannot be in D, but then it is an unselfish number and therefore must be in D, providing the contradiction.

If D is empty then every natural number is selfish and must map to a set that contains the natural number, and in this case no natural number can map to the in P(N) since the empty set contains no elements and, hence, includes no natural number.

This proves that the of the sets N and P(N) cannot be the same. Since P(N) contains N it has the larger uncountable cardinality (often |N| denoted as 1 = 2 where |N| = 0).

Infinite Infinities

The Power set of the Power set of N, and the Power set of that Power set, and so on indefinitely result in ever larger and larger sets. So, there can be infinite sizes of infinite sets denoted by the infinite sequence of aleph

numbers 0, 1, 2 …

Inconsistencies in the Proof of Cantor’s Theorem

To show one-to-one correspondence, if it exists, between the set N and its Power set P(N) is difficult enough. To prove that there cannot be one-to-one correspondence between these two sets should be an order of magnitude harder! How was Cantor able to establish such a difficult proposition so concisely? Did the proof involve some artful, innovative argument?

Not quite! Cantor’s proof relies on a number of questionable assumptions. Here is a list.

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Why does the Power set P(N) contain the empty set while the set N does not?

A set is defined as a fixed collection of distinct objects. S = {3, 5} is an example of a set that contains two different objects. A subset of a set includes one or more elements of that set and no other elements. The of {3, 5} should be the two sets {3} and {5}. One could technically count the entire set {3, 5} as a subset of itself.

The empty set denoted as {} is defined as the unique set with no elements.

 The empty set is inconsistent with both a set and a subset.

If there is no element, there is no set (or a subset of another set). In fact, the empty set used to be called the null set which denotes that there is no set.

The Power set of a set S, denoted as P(S), used to be the set of all subsets of S. This definition was later changed to include the empty set and the set S itself in addition to the subsets of S. With this revision, if S = {3, 5} then P(S) = {{}, {3}, {5}, {3, 5}}.

 Is the empty set forcibly included in the Power set to keep the proof of  Cantor’s theorem whole?

The Power set of the Power set of S, denoted as P(P(S)), includes the following objects:

The empty set {} Subsets of size 1 {{}} {3} {5} {3, 5} Subsets of size 2 {{}, {3}} {{}, {5}} {{}, {3, 5}} {{3}, {5}} {{3}, {3, 5}} {{5}, {3, 5}} Subsets of size 3 {{}, {3}, {5}} {{}, {3}, {3, 5}} {{}, {5}, {3, 5}} {{3}, {5}, {3, 5}} The set P(S) {{}, {3}, {5}, {3, 5}}

It has elements such as {} and {{}} which are considered to be distinct. The latter is just a set that contains the empty set. Can two “things” that both contain “nothing” be distinct? On the other hand, the empty set was characterized as unique, as one empty set is no different than another empty set. The concept becomes even more weird!

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The Power sets can be formed recursively, without an end, eventually leading to infinite elements such as

{}, {{}}, {{{}}}, {{{{}}}}, {{{{{}}}}}, …

All of which are a set that contains a set that contains a set that contains a set … that contains the empty set. Not a single element of the original set is present in any of these objects. Infinite distinct entities have been created out of nothing (the empty set). This theory is bizarre!!

 Cantor’s theorem breaks down if the empty set is not in P(N).

The arbitrary inclusion of the empty set in P(N) ensures that the set D of all unselfish elements cannot be empty (or there is a contradiction with the one-to- one correspondence between N and P(N) assumed at the beginning of the proof). This is the key step that enables such a simple proof of Cantor’s theorem! If D is empty, then the remaining problem is still to show there is no one-to-one correspondence between N and P(N) – the same problem as before!

The proof also breaks down if the empty set is included in the set N itself.

It defies common sense that the addition of just one more element makes the size of a set increase to a higher order of infinity. Surely, if the theorem is indeed true, there must be a way to prove it without the controversial insertion of the empty set in the Power set.

Does the Power Set of an Infinite Set Exist?

Does P(N) exist? Mathematics does not provide an existence proof. Instead, it relies on the axiom of Power set which stipulates that the Power set exists for every set, finite or infinite.

An axiom is a fancy name for an assumption. No infinite set (or quantity) can be constructed in reality. Simply assuming the existence of an infinite set and also that of the Power set of an infinite set are blatant violations of the concept of an axiom. It is no different than a religion assuming the existence of a God or many Gods, and then extending it further. Now the Gods must themselves have their own Gods, and on and on.

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Does the notion of selfish/unselfish elements apply to the empty set?

Can any condition involving elements apply to a set with no elements in it? How does one determine if the condition is satisfied or not?

If the empty set can be defined as a set with no elements then, analogously, shouldn’t there be a mapping with no elements involved in the mapping? If no natural number maps to a set that contains no natural number can such an association be considered a valid pairing (of no natural numbers) and called selfish? If yes, then the set D can be empty.

Does the set of all unselfish elements D exist?

Cantor’s original set theory resulted in many paradoxes and is considered naïve. Excerpt from the Wikipedia page “Russel’s Paradox” According to , 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.

To rule out the likes of Russell’s paradox the ZFC, for any property, explicitly disallows that there is a set of all things satisfying that property. The ZFC cannot have it both ways. If Russell’s paradox is not valid then it is inconsistent that the set of all unselfish numbers D is assumed to exist without providing an existence proof.

It would be crazy if there is a set of all things satisfying some properties but not others. Mathematics cannot be based on such inconsistent concepts and arbitrary rules!

The ZFC does contain the axiom of infinity and also the axiom of power set. Infinite sets are imaginary to begin with. The existence of the power set of an infinite set requires a further flight of fantasy. How does one show it exists? Making it an axiom stops all questions.

There is no reason to further debate these inconsistencies. The next section makes it all moot.

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Why Cantor’s Theorem is Ultimately Meaningless

Cantor’s theorem turns out to be meaningless for finite sets in that there is a hidden assumption in the axiom of Power set which is equivalent to the theorem’s conclusion. Moreover, this assumption is undefined when applied to an infinite set.

 Cantor’s theorem is simply presuming the ultimate result for a finite set  and it doesn’t even apply to an infinite set!

The definition of a set from finite set theory is all that is required to demonstrate this.

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

For example, in the set S = {1, 2, 3} each of the three unique objects “1”, “2” and “3” occurs just once in the set.

The cardinality of the of two sets isn’t the sum of the cardinalities of the two sets. The number of elements in the intersection of the two sets must be subtracted from the sum to enforce the condition there is a single occurrence of each element in the union.

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

The axiom of Power set starts with the existence of just (a single copy of) a set S – this alone is sufficient for its Power set P(S) to exist.

If S = {earth, mars} then its Power set P(S) = { {}, {earth}, {mars}, {earth, mars} }.

This Power set contains three real subsets of S and the empty set. Given just the set S any of its subsets can be formed one at a time. However, the three real subsets of S in the Power set P(S) cannot all exist simultaneously as a “completed totality”!

Only disjoint subsets of a set can exist at the same time.

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The subsets {earth} and {earth, mars} cannot exist concurrently. That would require two earths a condition not satisfied with just one copy of S.

The existence of P(S) requires two earths and two mars or two instances of the set S!

And, in general

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

Proof: By induction on n.

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

Proof: By induction on n, the sum of the cardinalities of all subsets in the Power set of a set with n elements turns out to be precisely n2n-1. It is the exact product of the cardinality n of the original set and the 2n-1 instances of the original set needed to construct the Power set.

The count of all objects involved in the formation of the Power set remains unchanged and equal to n2n-1. The Power set creates no net increase in cardinality.

If a Power set with 2n elements can be formed from a set with just n elements it would likely be a result of magic.

On the other hand, if 2n-1 copies of the original set of size n are required then the formation is revealed to be what it really is – a rearrangement. 2n-1 copies of a set of n distinct elements are rearranged to form 2n distinct subsets in the Power set of various sizes ranging from 0 to n. If the empty set is not included in the Power set, as it should not be, then the subset sizes range from 1 to n. There are more (sub)sets in the Power set (than the original sets at the start), but they are of smaller sizes, so it is an expected outcome and nothing profound!

Since finite sets are real it is possible to create explicit examples to demonstrate all valid concepts and even to negate other notions that are suspect. This is not possible with infinite sets as they are all imaginary. So, can one simply assume that the Power set of an infinite set exists as is done in the ZFC?

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 The Power set of an infinite set may not even exist.

The Power set of N can exist only if there are 2 to the power of infinity (minus 1) copies of the infinite set N to start. Such operations with infinite quantities are not allowed in mathematics since infinity is defined as “not a real number”. If the Power set of N exists it is equivalent to the supposition that the “quantity” 2 to the power of |N| (minus one) exists as the starting |N| condition. The uncountable cardinality is denoted as 1 = 2 .

 The assumption (or axiom) that the Power set of N exists has presumed  that a higher order of infinity already exists!

Infinite set theory is profoundly wrong. Many strange entities are assumed to be sets in direct violation of the very basic concept of a set.

A finite set is just a fixed collection of distinct objects.

It cannot be that a set of all things can be constructed for some properties but not for others, and Russel’s set does not exist while the set D does. Instead, entities with rules governing membership are sets only if

The rules or conditions or notation involved in the formation of the entity are such that they can be fully exercised prior to the formation of the set, and all elements of the set exactly determined up front.

Neither Russel’s set nor the set D meet this criterion. The entity [1, 2, 3 … 10] can be expanded to yield the entire collection of objects [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] and is a valid set. The and the Field axioms that form the basis of the “set” of all natural numbers N and the “set” of all real numbers R cannot be resolved entirely. They result in an unending list of numbers and cannot become a fixed collection of distinct objects. The notion of an actual infinity cannot simply assume N and R are completed objects!

 Stating a self-contradictory idea as an axiom does not make it true!!

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Conclusions

How can such an awe-inspiring result like Cantor’s theorem turn out to be meaningless for finite sets and undefined for infinite sets?

How can the axiom of power set be so 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 .”

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

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

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

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

The five books listed in the references explain exactly why.

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

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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 has been misapplied.

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

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

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

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

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

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

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

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

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

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

A Book by Wolfgang Mückenheim

Transfinity A Source Book, (April 2018, previous version March 2017)

This book presents the theory of actual infinity and of transfinite set theory. The attitude of the founder of transfinite set theory, Georg Cantor, is illuminated with respect to sciences and religion by his quotes as well as those of his followers in chapter IV. The set of applications of set theory are also summarized. Quotes expressing a skeptical attitude against transfinity and addressing questionable points of current mathematics are collected in chapter V. The critique is scrutinized in chapter VI, the main part of this source book. It contains over 100 arguments against actual infinity – from doubtful aspects to clear contradictions.

Alternate Titles: Why Cantor’s theorem is wrong Why Cantor’s theorem is inconsistent The Flaw in Cantor’s theorem The Mistake in Cantor’s theorem

Keywords: Georg Cantor, Infinite Set Theory, Cantor’s Infinite Set Theory, Cantor’s theorem, Power set, Empty Set, Null Set,

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