Alex A. Zenkin

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• ON THE LOGIC OF "PLAUSIBLE META-MATHEMATICAL FALLACIES

A.A. ZENKIN
Computing Center of the Russian Academy of Sciences

ON THE LOGIC OF "PLAUSIBLE"
META-MATHEMATICAL FALLACIES.

	Abstract. - This paper examines some new aspects of the logic of Cantor's proof of the uncountability of the continuum using the diagonal method.

	It is proven for the first time that the key point of Cantor's proof of the uncountability of the continuum by the Reductio ad Absurdum (RAA) method is the explicit use of the counter-example method.
	Let's consider Cantor's traditional proof [1,2].
	CANTOR'S THEOREM (1890). {A:} The set X=[0,1] is uncountable.
	PROOF (by RAA method). Suppose that {A:} X is countable. Then there exists a 1-1 correspondence between the elements of sets X and N={1,2,3,…}. Let {B:}

	x1, x2, x3, … 		(1)

- be some enumeration (list) of all real numbers (hereinafter - r.n.) from X. In this case, using his famous diagonal method, Cantor constructs a new r.n., say, y, which, by construction, is different from every r.n. in list (1), i.e., y  (1). Consequently, {B:} list (1) does not contain all r.n. from X. According to Cantor's version of the RAA method (see below), the obtained contradiction between B and B in the form of a deductive inference [B  B] proves that assumption A is false. Consequently, A is true. Q.E.D.
	From the point of view of classical logic, Cantor's r.n. y is a counter-example, refuting the RAA assumption about the existence of an enumeration (1) of all r.n. from X. As is known, a single counter-example is sufficient to refute a general statement, and the question of the number of elements (cardinality) of the set of all possible counter-examples is an independent and separate task. In other words, in Cantor's RAA proof of the uncountability of the continuum, the quantitative characteristics (cardinality) of the continuum are not used and play no role.
	An attempt to directly define the cardinality of the continuum within the framework of Cantor's RAA proof leads to a non-finite process (here B="list (1) contains all r.n. from X"):

B  B  B  B  B  B  B  . . . ,

which excludes the possibility of refuting the RAA assumption about the countability of the continuum, i.e., Cantor's statement about the uncountability of the continuum becomes unprovable. It is proven that this non-finite process represents a new paradox (of the "Liar" type) in Cantor's set theory [3-5].
	It is also proven that if, in the assumption of Cantor's RAA proof of the countability of the continuum, all elements of the set N={1,2,3, . . .} are used to index the r.n. in list (1), then Cantor's conclusion that the cardinality of the continuum is greater than the cardinality of N is "impeccable"; however, if elements of another countable set, say, {2,4,6, . . .}, are used to index the r.n., then within the framework of the same Cantor's RAA proof, a conclusion directly opposite to Cantor's is obtained: the cardinality of set N is not less than the cardinality of the continuum. This means that the cardinality of the continuum essentially depends on the indexing of its elements in sequence (1), which is an absurdity from the point of view of modern axiomatic set theory [1, 3-7].
	The facts presented indicate the fatal inconsistency of Cantor's set theory and illustrate the basic meta-mathematical paradigm: "from a contradiction (for example, of the "Liar" type) anything follows," and, in particular, a rather "plausible" conclusion about the uncountability of the continuum.

References.

	1. G. Cantor, Works on Set Theory. - M.: Nauka, 1985.
	2. P.S. Aleksandrov, Introduction to General Set Theory and Functions. - Moscow-Leningrad: Gostekhizdat, 1948.
	3. A.A. Zenkin, The Principle of Time Separation and Analysis of One Class of Quasifinite Plausible Reasoning (on the example of G. Cantor's theorem on uncountability). - Doklady Akademii Nauk, vol. 356, No. 6, 733-735 (1997).
	4. A.A. Zenkin, Georg Cantor's Error. - Voprosy Filosofii, 2000, No. 2, 165-168.
	5. A.A. Zenkin, A New Approach to the Analysis of the Problem of Paradoxes. - Voprosy Filosofii, 2000, No. 10, 79-90.
	6. A.A. Zenkin, "Infinitum Actu Non Datur". - Voprosy Filosofii, 2001, No. 9, 157-169.
	7. A.A. Zenkin, Scientific Intuition Of Genii Against Mytho-"Logic" Of Transfinite Cantor's Paradise. International Symposium - Philosophical Insights into Logic and Mathematics, 2002, Nancy, France. Proceedings, pp. 141-148.