Fuzzy sets within Finitely Supported Mathematics

The classical theory of fuzzy sets is extended to a recently developed framework named Finitely Supported Mathematics in order to characterize fuzzy sets over infinite universes in a finitary manner by involving the concept of “finite support”. We prove some algebraic properties of the new fuzzy sets within Finitely Supported Mathematics (including some that cannot be obtained in Zermelo-Fraenkel mathematics), and introduce operations and extension principles over these fuzzy sets. We introduce a specific (infinite) membership-degree association, and connect it to the notions of invariant complete lattices and invariant monoids in Finitely Supported Mathematics.