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 cardinality of power-set, |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 element of P(X) not belonging to a mapping of X onto P(X)) is sufficient in order 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 number 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 relation 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 subset 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 bijection 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 set theory [1-4]. DEFINITION-1. A set, Z, is infinite iff it’s equivalent to its own subset. LEMMA-1. If a difference between numbers 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 group 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.