Функциональные алгебраические модели для неклассической теории множеств.

Abstract

It is well known that a general method of investigation of nonstandard logics is the construction of such algebraic models in which true valuations are elements of some algebra. For example, if we investigate intuitionistic logic, we consider pseudoboole (or Heyting) algebra. In 1979 A.Dragalin suggested a general construction for predicate logic (of course, imuitionistic primarily). This construction is the uniform interpretation of intuitionistic arithmetic models (of realizability types).

In the present work Dragalins method raised for such nonpredicative theories as a set theory with extensionality axiom and also uniform approach to construction of set theory realizability type models is given (we add to the set theory an axiom of double completion, as such set theory is equiconsistent to classical ZF).

In the first part of our short work Dragalin s method is clarified and then (second part) we give analogical construction for the set theory. The main result states: if F is a functional algebraic model for set theory language then valuations of all axioms of predicate logic and standard axioms of set theory are equal to the function f from F which is unitconstant.