A graded approach to cardinal theory of finite fuzzy sets, part I: Graded equipollence

In this article, we propose a fuzzy class relation assigning to each pair of finite fuzzy sets a degree to which they are equipollent, which indicates that they have the same number of elements. The concepts of fuzzy sets and fuzzy classes in the class of all sets (in ZFC) are introduced, and several standard relations and constructions, such as the fuzzy power set and exponentiation, are defined. A functional approach to the cardinal theory of finite fuzzy sets based on graded equipollence is shown, and a relation to generalized cardinals and Wygralak's cardinal theory of finite fuzzy sets defined over triangular norms is demonstrated.