The formal-logical analysis of the foundation of set theory Kalanov Temur Z.* Home of Physical Problems, Yozuvchilar (Pisatelskaya) 6a, 100200, Tashkent, Uzbekistan *Corresponding Author: Temur Z. Kalanov, Home of Physical Problems, Yozuvchilar (Pisatelskaya) 6a, 100200, Tashkent, Uzbekistan. E-mail: tzk_uz@yahoo.com, t.z.kalanov@mail.ru, t.z.kalanov@rambler.ru
MSC: 00A05, 00A30, 00A35, 03A10, 03B44, 03B80, 03B99, 03F99, 03F03, 03E75, 03F03, 03E20, 08C99, 54A99, 97E20, 97E30, 97E50, 97E60, 97A99. Online published on 16 January, 2018. Abstract The critical analysis of the foundation of set theory is proposed. The unity of formal logic and rational dialectics is the correct methodological basis of the analysis. The analysis leads to the following results: (1) the mathematical concept of set should be analyzed on the basis of the formal-logical clauses “Definition of concept”, “Logical class”, “Division of concept”, “Basis of division”, “Rules of division”; (2) the standard mathematical theory of sets is an erroneous theory because it does not contain definition of the concept “element (object) of set”; (3) the concept of empty set (class) is a meaningless, erroneous, and inadmissible one because the definition of the concept “empty set (class)” contradicts the definition of the logical class. (If the set (class) does not contain a single element (object), then there is no feature (sign) of the element (object). This implies that the concept of empty set (class) has no content and volume (scope). Therefore, this concept is inadmissible one); (4) the standard mathematical operations of union, intersection and difference of sets (classes) are meaningless, erroneous and inadmissible operations because they do not satisfy the following formal-logical condition: every separate element (object) of the set (class) must be in only one some set (class) and cannot be in two sets (classes). Thus, the results of formal-logical analysis prove that the standard mathematical theory of sets is an erroneous theory because it does not satisfy the criterion of truth. Top Keywords Set theory, Applications of set theory, Foundations of mathematics, General mathematics, General algebra, General topology, Methodology of mathematics; Logic, General logic, Temporal logic, Formal logic, Proof theory, Philosophy of mathematics, Mathematics education, Logic in the philosophy of science, Dialectics. Top |