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This paper presents an axiomatic set theory FST ('Fuzzy Set Theory'), as a first-order theory within the framework of fuzzy logic in the style of [4]. In the classical ZFC, we use a construction similar to that of a Boolean-valued universe--over an algebra of truth values of the logic we use--to show the nontriviality of FST. We give the axioms of FST. Finally we show that FST interprets ZF.