Cardinality of Fuzzy Sets and Accumulation of Small Membership

We describe an intuitive and practically significant empirical phenomenon that relates to the concept of cardinality of a fuzzy set, namely, an excessive accumulation of small degrees of membership. We argue and demonstrate by examples that the present notions of cardinality do not take this phenomenon into account properly and may thus prove insufficient in applications. We propose a new concept of cardinality, generalizing the well-known Zadeh's sigma count, demonstrate using both intuitive and technical examples that it alleviates the insufficiency of the existing ones, and provide a theoretical analysis of this concept. We also propose topics for future theoretical and empirical research.