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Home , Bijection, Cardinality

CANTOR’S DIAGONAL ARGUMENT: A NEW ASPECT.

Alexander.A.Zenkin ( [email protected] )

Dorodnitsyn Computing Center of the Russian Academy of Sciences.

Abstract. – In the paper, Cantor’s diagonal proof of the theorem about the of power-, |X| < |P(X)|, is analyzed. It is shown first that a key point of the proof is an explicit usage of the counter-example method. It means that an only counter-example (Cantor’s new of P(X) not belonging to a mapping of X onto P(X)) is sufficient in to formally disprove a common statement (the assumption of Cantor’s proof that there is a mapping of X onto P(X) including all elements from P(X)), but a total of all possible counter-examples (a cardinality of P(X)) plays no role in such disproof. In addition Cantor’s conclusion in the form |X| < |P(X)| is deduced from the fact that the difference between infinite sets, P(X) and X, amounts to one element, that is such conclusion contradicts fatally the main property of infinite sets. So, it takes place the following unique situation: the formal logic of Cantor’s proof is unobjectionable, but the proof itself has no to and does not use quantitative properties, i.e., a number of elements or a cardinality, of the set, P(X). It is proved as well that if to suppose that a set of all possible Cantor’s counter-examples is infinite, then the Cantor argument leads to an infinite “reasoning” which does not allow to disprove the assumption, |X| = |P(X)|, i.e., makes Cantor’s statement, |X| < |P(X)|, unprovable within the framework of just traditional Cantor’s proof and from the point of view of classical mathematics.

Consider the Cantor theorem about the cardinality of a power-set [1,2] and its traditional diagonal proof [3]. Here and further P(X) is a power-set and |X| is a cardinality of an arbitrary set X, and, for short, RAA = Reductio and Absurdum, CDM = Cantor’s Diagonal Method, AD-element = Anti- Diagonal element generated by the CDM-application.

CANTOR’S THEOREM-1 (1890). |X| < |P(X)|. PROOF-1 (by RAA-method). It’s obvious that |X| £ |P(X)|. Assume that j maps X onto P(X). Define a new of X as follows: X*={x Î X | x Ï j(x)}. Then X* Í X, and if X* = j(y) for some y Î X, then yÎ X* ® y Ï j(y) ® y Ï X* and y Ï X* ® y Î j(y) ® y Î X*. The last is impossible. Q.E.D.

As Esenin-Volpin notes rightly (see his commentary in [3], p.172]), the main stage of Cantor’s proof is a proof of the statement that “for any X there is not a of X onto P(X)”. Unfortunately, this very important remark of Esenin-Volpin was not appreciated at its true value and did not have a continuation. However this short comment leads to very important consequences concerning a true understanding of a logical nature of Cantor’s diagonal argument.

To begin an analysis of the Cantor proof, remind of some fundamental statements of modern [1-4].

DEFINITION-1. A set, Z, is infinite iff it’s equivalent to its own subset. LEMMA-1. If a difference between of elements of two infinite sets, say Z1 and Z2 , is finite, then the sets, Z1 and Z2 , are equivalent, i.e., |Z1| = |Z2|. DEFINITION-2. Two infinite sets, say Z1 and Z2, are equivalent, i.e., |Z1| = |Z2|, iff there is a 1-1-correspondence between Z1 and Z2 . LEMMA-2. If there is not a 1-1-correspondence between elements of sets, Z1 and Z2, then

- 1 - the sets Z1 and Z2 are not equivalent, i.e., |Z1| ¹ |Z2|.

Now, taking into account the Esenin-Volpin comment, rewrite Cantor’s Theorem-1 and its traditional proof in the following equivalent form. Remark that we shall speak “an element YÎP(X) belongs to the mapping j (and write YÎj) if there is xÎX such that Y=j(x)”.

CANTOR’S THEOREM-2 (1873, 1890). There is not a bijection between X and P(X) for any X. PROOF-2 (by RAA-method). Assume that there is a bijection j that maps X onto P(X). Following Cantor, define a new subset of X: X*={x Î X | x Ï j(x)}. Then X*ÎP(X), but "YÎj[X* ¹ Y], i.e., X*Ïj. Indeed, if X* = j(y) for some y Î X, then y Î X* ® y Ï j(y) ® y Ï X* and y Ï X* ® y Î j(y) ® y Î X*. The last is impossible. From the proven fact X*Ïj, it follows that the given mapping j is not a 1-1-correspondence between X and P(X). Consequently, by virtue of the arbitrariness of j, there is not a bijection between X and P(X) for any X. Q.E.D. CANTOR’S THEOREM-3. |X| < |P(X)|. PROOF-3. From Theorem-2, Lemma-2, and the obvious inequality |X| £ |P(X)|, it follows that |X| < |P(X)|.

So, the idea, prompted by the Esenin-Volpin comment, consists in an explicit separation of a logical (Cantor’s Theorem-2) and set-theoretical (Cantor’s Theorem-3) aspects of the traditional Cantor diagonal argument. A comparison of the logic of Cantor’s Proof-2 with the logical scheme of the classical method of counter-example allows to reveal a unique meta-mathematical fact that a key point of Cantor’s diagonal argument is an explicit usage just of the counter- example method. Indeed, consider a logical scheme of classical counter-example method. 1. There is a common hypothetical statement, "x P(x), xÎD, where D is a set of all possible values of x for which the predicate, P(x), is defined. 2. By means of a searching in D (e.g., by Monte Carlo method), the fact is established (if any) that $x*ÎD ØP(x*). 3. From this only fact it follows that Ø["xÎD P(x)].

CONSEQUENCE-1. In order to disprove a common statement, "x P(x), an only counter- example x*, for which ØP(x*), is sufficient. CONSEQUENCE-2. The cardinality of a set of all possible counter-examples, x, for which ØP(x), is not taken into account and is not used within the framework of the counter-example method.

Consider one of the most bright examples of a real application of counter-example method in mathematics.

EXAMPLE-1. In the XVIII century great Euler formulated the following quite plausible common statement. EULER’S CONJECTURE. For any exponent r ³ 3 the Diophantine equation, r r r r r n = n1 + n2 + n3 + … + ns , has no solutions in natural numbers if s < r. By s=2, Euler’s Hypothesis entails Last Fermat Theorem. During about 200 years nobody was able either to prove or disprove the Hypothesis. And only in 1967 a of American mathematicians with a power computer revealed ... only one counter-example [5], 1445 = 275 + 845 + 1105 + 1335 ,

- 2 - where r = 5, s = 4, i.e. s < r. This only counter-example disproved Euler’s Hypothesis forever. So, repeat once again, in order to disprove a common statement an only counter-example is sufficient (Consequence-1, above). And the fact that a set of such counter-examples may be (in reality) infinite plays no role in such disproof (Consequence-2, above). In other words, the disproof of a common statement by means of a given counter-example and the question about a factual number of such counter-examples, i.e., about the cardinality of the set of all possible counter-examples, are absolutely different, independent problems.

Now consider another example of a usage of the counter-example method.

EXAMPLE-2. Within the framework of traditional Cantor’s diagonal proof of the Theorem- 1, there is the following fragment. According to the assumption of Cantor’s proof, there is a bijection j: X ® P(X), i.e., {B:} all elements of P(X) belong to the given j. Then Cantor defines a new object X*={x Î X | x Ï j(x)} such that X*ÎP(X), but X*Ïj. It means that $X*ÎP(X)[X*Ïj], i.e., {ØB:} not all elements of P(X) belong to the given j. So, from the point of view of classical logic, Cantor’s AD-element, X*, is a counter-example disproving the common statement B.

These two examples show that there are two different versions of the counter-example method: in the first version (call it a classical one) a counter-example is searched for in a set of all possible realizations of a common hypothesis; in the second version (call it a CDM-version) a counter-example is algorithmically deduced from the hypothesis which this counter-example must disprove.

Remark, that the CDM-version is a distinctive feature just of Cantor’s diagonal argument and can be formulated in the following common form [10].

STATEMENT-1. The famous Cantor Diagonal Method (in its any meta-mathematical realizations) is a special case of the Counter-Example Method where a counter-example itself is not searched for in a set of all possible realizations of a given common statement, but is algorithmically deduced from the given common statement, which this counter-example must disprove.

Now we shall prove some theorems elucidating some “hidden” logical peculiarities of the Cantor’s diagonal argument just from the set-theoretical point of view.

THEOREM 1. Cantor’s conclusion |X| < |P(X)| contradicts Lemma 1 [6,7,10,11]. PROOF (by RAA-method). Following Cantor, assume that j maps X onto P(X), i.e., |X| = |P(X)|. Represent the set P(X) as the sum, P(X) = P1+P2, where P1 is a set of all elements from P(X), which really belong to j, and P2 is a to P1 in P(X), i.e., P2 = P(X) – P1. It’s obvious that, by virtue of the RAA-assumption, |P1|=|X| and P2 is an . For the given j Cantor defines X*={x Î X | x Ï j(x)} such that X*ÎP(X), but X*Ïj, i.e., X*Ï P1, . It means that X*Î P2, i.e., Cantor proves that P2 is not an empty set. As said above, a key point and a specific peculiarity of Cantor’s RAA-proof is an explicit usage of the counter-example method, since, from the point of view of classical logic and classical mathematics, Cantor’s anti-diagonal AD-element X* is a counter-example, disproving the common statement B = «a given mapping j includes all elements from P(X)». From the proven falsity of the statement B (just by means of the CDM-version of the counter-example method) it follows (by classical modus tollens rule) that the RAA-assumption, |X| = |P(X)|, is false. By virtue of Lemma-2, together with the obvious inequality, |X| £ |P(X)|, it leads to the finale Cantor statement |X| < |P(X)|. - 3 - So, from the point of view of a formal logic, Cantor’s RAA-proof is unimpeachable and irrefutable. However, it’s obvious that for a given (fixed) mapping j, the Cantor’s diagonal method is able to produce the only, unique element X* of the set P(X), not belonging to the given j, i.e., |P2| = 1. It means that Cantor’s conclusion |X|<|P(X)| is based on the fact that infinite set P(X) has only one element greater than the X, i.e., |P(X)| - |X| =1. It’s obvious that such set theoretical “ground” for Cantor’s conclusion |X|<|P(X)| contradicts fatally to the set- theoretical Lemma 1, according to which, the |P(X)| - |X| =1 entails |X| = |P(X)|. Q.E.D.

Now, we shall consider the case when the application of CDM to j is, supposedly, able to generate an infinite set of new counter-examples. In such a case the following statement takes place.

THEOREM 2. The Cantor inequality |X|<|P(X)| is unprovable [8 - 11] PROOF. The only reason to state that |X|<|P(X)| is Cantor’s Theorem-1 above. Therefore in order to prove our Theorem 2 it’s sufficient to prove that traditional Cantor’s diagonal proof does not prove the statement |X|<|P(X)|. Toward this end, consider again the traditional Cantor’s proof. CANTOR’S THEOREM. |X| < |P(X)|. PROOF-1 (by RAA-method). Assume that j maps X onto P(X), i.e., |X| = |P(X)|. Represent the set P(X) as the sum, P(X) = P11+P12, where P11 is a set of all elements from P(X), which really belong to j, and P12 is a complement to P11 in P(X), i.e., P12 = P(X) – P11. It’s obvious that, by virtue of the RAA-assumption, |P11|=|X|, and P12 is an empty set. For the given j Cantor defines X*={x Î X | x Ï j(x)} which does not belong to j, and proves that P12 is not empty. But now we shall admit that changing the initial mapping j (without a change of a number of elements of P(X) belonging to the initial mapping j), Cantor’s diagonal procedure is able to produce an infinite set P12 of new AD-elements from P(X), not belonging to the initial mapping j, i.e., not belonging to P11. The last means that now the veritable cardinality of P(X) is defined by and is equal to the cardinality of the complement P12, i.e., |P(X)| = |P12|. The following two cases are possible. (i) |P12| = |X|. If that is so, then |P(X)| = |P11 + P12| =|X|, and therefore to disprove the RAA- assumption |X| = |P(X)| is impossible from the point of view just of axiomatic set theory. (ii) |P12| > |X|. However, since Cantor’s proof so far is not completed, the very existence of a cardinality which is greater than |X| is so far not proven, and therefore a hypothetical statement, |P12| > |X|, must be proved. It can be done by the only way – by means of the CDM, i.e., one must now prove the initial Cantor’s Theorem with the new symbol P12 instead of the old symbol P(X).

CANTOR’S THEOREM. |X| < |P12|. PROOF-2. Assume that j1 maps X onto P12, i.e., |X| = |P12|. Suppose that the application of the CDM to j1, produces an infinite set of elements from P12, which don’t belong to the j1. Represent the set P12 as a sum P12 = P21 + P22, where P21 is a set of elements of P12 really included in the j1, but the set P22 is a complement to P21 in P12, i.e., P22 = P12 – P21. It’s obvious that |P21|=|X|, and the complement P22 contains all AD-elements, which can be produced by the CDM-application to the j1. The last means that now the veritable cardinality of P12 is defined by and is equal to the cardinality of the complement P22, i.e., |P12| = |P22|. Consider the following two cases. (i) |P22| = |X|. If that is so, then |P12| = |P21 + P22| =|X|, and therefore it’s impossible to disprove the RAA-assumption, |X| = |P12|, from the point of view just of axiomatic set theory. (ii) |P22| > |X|. However, since Cantor’s proof so far is not completed the very existence of a cardinality which is greater than |X| is so far not proven, and therefore a hypothetical statement, |P22| > |X|, must be proved. It can be done by the only way – by means of the CDM, i.e., one must now prove the initial Cantor’s Theorem with the new symbol P22 instead of the old symbol P12.

- 4 - And so on ad infinitum. Thus, the traditional Cantor “proof” of the Theorem about the cardinality of a power-set, from the set-theoretical point of view (!), either isn’t able to disprove the RAA-assumption, |X| = |P(X)|, or is reduced to the infinite system of “nested” (“embedded”) proofs of the initial Cantor Theorem by the sequential replacement of the initial symbol P(X) by symbols P12, P22, P32, …, that is to the following non-finite, tautological, and quite senseless “reasoning” (here Di = «it needs to prove that |X| < |Pi2| »):

D1 ® D2 ® D3 ® … (*)

It’s obvious, that until the potentially infinite (obviously, countable) “reasoning” (*) is finished, the initial RAA-assumption |X| = |P(X)| of Cantor’s RAA-proof is irrefutable from the set-theoretical points of view, and, consequently, the Cantor statement |X| < |P(X)| is unprovable. Q.E.D.

So, the Cantor strict inequality |P(X)| > |X| is not provable. By virtue of the obvious inequality |P(X)| ³ |X|, it means that |P(X)| = |X|. However, in this case, unique, specific properties of CDM allow to prove the following, quite unexpected, ‘intuitionistic’ statement.

THEOREM 3. If X is equivalent to P(X), then there is not a rule (algorithm) to produce a 1- 1-correspondence between elements of X and P(X). PROOF. Assume that |X|=|P(X)|, but there is a rule (algorithm) producing a 1-1- correspondence between elements of X and P(X), i.e., there is j, mapping X onto P(X) and containing all elements from P(X), i.e., "YÎP(X)[YÎj]. Applying CDM-algorithm to the mapping j, we shall construct a new element, X*={xÎX | xÏj(x)}, which doesn’t belong to j, i.e., $X*ÎP(X)[X*Ïj]. Contradiction. Q.E.D.

So, there are two legitimate reasons of the non-existence of a 1-1-correspondence between X and P(X): (i) |P(X)| > |X| - according to Cantor’s definition of the notion of the non-equivalency of sets (Lemma-2); (ii) |P(X)| = |X|, but there is not a rule (algorithm) to produce a real 1-1-correspondence between X and P(X). It is obvious that the case (ii) contradicts Cantor’s Definition-2 of the notion itself of the equivalency of infinite sets and Lemma-2. The fact demands a more careful analysis of foundations of Cantor’s “Study on Transfinitum”, but such analysis is not a point of this paper.

CONCLUSIONS.

1. For the first time the fact is revealed and explicitly formulated that the crucial point of Cantor’s RAA-proof of the power-set theorem, stating that |X| < |P(X)|, is an explicit usage of the counter-example method. It is shown that the famous Cantor Diagonal Method (in its any meta- mathematical realizations) is a special case of the Counter-Example Method where a counter- example itself is algorithmically deduced from the given common statement, which this counter- example must disprove. 2. The formal logic of Cantor’s RAA-proof seems to be unimpeachable and irrefutable. Indeed, from the point of view of the logic, an only counter-example (Cantor’s AD-element X*Ïj) is sufficient in order to disprove the RAA-assumption of Cantor’s diagonal proof that there is a bijection, j, between X and P(X). On the other hand, Cantor’s conclusion in the form |X| < |P(X)| is deduced from the fact that the difference between infinite sets, P(X) and X, amounts to one element, i.e., |P(X)| - |X| =1. It is obvious that this Cantor conclusion contradicts

- 5 - fatally the main property of infinite sets in the form of Lemma 1, according to which from the fact |P(X)| - |X| =1 it follows that |X| = |P(X)|, i.e., that the sets X and P(X) are equivalent. It means that there takes place the following unique (in all history of mathematics) situation: on the one hand, Cantor’s conclusion, |X| < |P(X)|, is unobjectionable from the point of view of formal logic, but, on the other hand, the diagonal proof of the conclusion is based on the fact which contradicts Lemma 1, i.e., the Cantor conclusion that |X| < |P(X)| is wrong from the point of view of modern axiomatic (‘non-naive’) set theory. The proof itself has no relation to and does not use quantitative properties, i.e., a number of elements or a cardinality, of the set, P(X). 3. It is proved that in a common case when a set of all possible Cantor’s counter-examples is, supposedly, infinite, the Cantor strict inequality, |X| < |P(X)|, becomes unprovable within the framework of traditional Cantor’s proof and from the point of view of classical mathematics.

REFERENCES.

[1] , Uber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen, Journal fur die reine und angewandte Mathematik, vol. 77 (1874), pp. 258–262. [2] Georg Cantor, Proceedings in Set Theory, Moscow, NAUKA, 1985 [3] Paul J.Cohen, Set Theory and the -Hypothesis, Stanford University, W.A.Benjamin, Inc., New York – Amsterdam, 1966, Chapter II, §4, Cardinal Numbers. [4] Stephen C.Kleene, Introduction to Metamathematics, D.Van. Nostrand Company, Inc., New York – Toronto, 1952. [5] Lander L.J., Parkin T.R. A counter example to Euler's sum of power conjecture, Math. Comp., 21, 101 - 103 (1967). [6] A.A.Zenkin, George Cantor's Mistake. - Voprosy Filosofii (Philosophy Problems), 2000, No. 2, pp. 165-168. http://www.com2com.ru/alexzen/papers/vf1/vf-rus.html http://www.com2com.ru/alexzen/papers/Cantor/Fatal_Mistake_of_Cantor.html, [7] A.A.Zenkin, "Infinitum Actu Non Datur". – Voprosy Filosofii (Philosophy Problems), 2001, No. 9, pp. 157-169 [8] A.A.Zenkin, Scientific Intuition Of Genii Against Mytho-‘Logic’ Of Transfinite Cantor's Paradise, Proceedings of International Symposium “Philosophical Insights into Logic and Mathematics (PILM-2002)”, Nancy, France, 2002, pp. 141-148. [9] A.A.Zenkin, On one reconstruction of Wittgenstein's objection against Cantor's Diagonal Argument, Proceedings of VII scientific Conference "Modern Logic: Problems of Theory, History, and Applications in Science", Sankt-Petersburg, 2002, pp. 320-323. [10] A.A.Zenkin, On Logic of “plausible” meta-mathematical misunderstandings, Proceedings of All-Russian Conference “Scientific Session of MEPI-2004”, Moscow Engineering & Physical Institute, 2004, pp. 182-183. [11] A.A.Zenkin, A priori logical assertions with a zero ontology, In the Collection “Mathematics and Experience”, (Prof. A.G.Barabashev, ed.), Moscow state University publishing house, 2004, pp. 423-434.

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