A set theory within fuzzy logic

Abstract

This paper proposes a possibility of developing an axiomatic set theory, as first-order theory within the framework of fuzzy logic in the style of Hajek's Basic fuzzy logic BL. In classical Zermelo-Fraenkel set theory, we use an analogy of the construction of a Boolean-valued universe-over a particular algebra of truth values-we show the nontriviality of our theory. We present a list of problems and research tasks.