THE PROBLEM OF SEMANTIC REDUNDANCY AND CERTAINTY OF THE CONTINUUM HYPOTHESIS IN AXIOMATIC FIRST AND SECOND ORDER SET THEORIES
V. V. Tselishchev
Institute of Philosophy and Law SB RAS, 6300909, Russia, Novosibirsk, st. Nikolaeva, 8
Keywords: семантическая избыточность, континуум-гипотеза, первопорядковая теория множеств, второпорядковая теория множеств, разрешимость, неформальная строгость, Semantic redundancy, the continuum hypothesis, first-order set theory, second-order set theory, decidability, informal rigor
Abstract
The purpose of this article is to explain the difficulties in solving some problems the set-theoretic nature by semantic redundancy of language for describing the sets, namely, in the Zermelo-Fraenkel system with the Axiom of Choice and second-order set theory. We describe two approaches to the problem by means of case of the continuum hypothesis, undecidable in ZFC system and decidable in an alternative ZFC2 set theory. It is shown that the basis of difference of interpretations lies in pre-theoretical concepts that allow semantic redundancy.
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