Functors of Boolean-Valued Analysis

Abstract

Owing to the transfer and maximum principles, various constructions of conventional mathematical practice are possible within a Boolean-valued universe. In particular, in such a model there are fields of real numbers, Banach spaces, differential operators, etc.. The objects presenting them serve, in a certain sense, as a nonstandard realizations of the initial mathematical constructions. Therefore, assuming that the model V^(B) is a nonstandard presentation of the mathematical world and taking into account the fact that V^(B) is constructed within the von Neumann universe, we are in a position as if we look inside the Boolean-valued world and see a standard presentation of nonstandard objects. While examining algebras B item-by-item, an observer sees a great number of modifications of the same idea encoded in a set-theoretic formula. It is comparing them with one another that comprises the method of studying the mathematical idea they are based on. Besides, one discovers that essentially different analytical objects are nothing but presentations of the same concept. This fact enables one to clarify the internal reasons accounting for many analogies, as well as facilitates the appearance of new opportunities to study old objects.