The aim of this paper is to overview our recent studies on mathematical foundation of fuzzy set theory, and to show that large parts of fuzzy set theory can be explained by our natural interpretation. With the natural interpretation of fuzzy set theory in a cumulative Heyting valued model for intuitionistic set theory, various notions and properties of fuzzy sets, fuzzy relations and fuzzy mappings can be acquired consistently.
As far as fuzzy set theory is considered as an extension of usual set theory, this interpretation seems to be most natural.