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A.A.Zenkin, AS TO LOGIC OF CANTOR’S DIAGONAL ARGUMENT (in English, in Russian ).
A.A.Zenkin, CANTOR’S DIAGONAL ARGUMENT: A NEW ASPECT. (in English, in Russian)
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 a 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 “implication” 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. A.A.Zenkin, "Scientific Counter-Revolution in Mathematics". - Nezavisimaya Gazeta on 19 July, 2000 (Independent Newspaper). Supplement "NG-Science", pp. 13. (http://science.ng.ru/magnum/2000-07-19/5_mathem.html ).