Definitions and presuppositions in set theory

The article presents a study of ontological presuppositions that lie in the ground of various realizations of set theory. The author claims that set theory is the universal language that stipulates the progress in mathematics in the last 150 years and set-theoretical paradoxes are not defects. First, the author shows that the equivalence sign in formulas defining sets means not the equivalence relation but rather the definition operator, so the rules of definition could be applied to such kinds of formulas and related expressions, and if all rules are complied with, there is no way to get any set-theoretical paradox. Next, the author examines various realizations of set theory, like naive Cantorian set theory, Bolzano’s theory of Inbegriff, Russell’s type theory, Zermelo-Fraenkel axiomatization, etc., and shows hidden presuppositions that lie at the base of each set theory and deny the possibility of paradoxes. Cantor’s and Bolzano’s theories use specific ontological presuppositions, and other theories use restrictions provided by language to confine the language of set theory and deny paradoxes. So set-theoretical paradoxes are external for those theories and could be provided only if such restrictions are ignored. The plausibility of paradoxical propositions is not the lack of the set theory language but the evidence of its expressive power, which could be odd in some cases. All described theories could work only with the decidable set when the language of set theory could provide propositions about the undecidable set also. In this case, the set-theoretical paradoxes are no less than demonstrations of the undecidability of certain sets. There are set theories, like axiomatization of von Neumann-Bernays-Godel, Zadeh’s fuzzy sets, and Vopenka’s alternative set theory, that provide instruments to deal with such nonstandard sets. For such extensions of the classic set theory it is necessary to modify not only its conceptual construct but also the logical system that lies at its base, for example, loosening restrictions on contradictions. The author declares no conflicts of interests.

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