GENERALIZED CARDINAL NUMBERS AND OPERATIONS ON THEM

In this paper we present a cardinality theory for so-called vaguely defined objects which are mild generalizations of fuzzy sets, obtained by introducing lower and upper approximations of the membership functions. The theory makes use of the sentential calculus in the infinite-valued Lukasiewicz logic, and can be applied to fuzzy sets with arbitrary supports, (some kinds of) twofold fuzzy sets, partial sets, etc. The notion of equipotency of vaguely defined objects is studied in detail. The resulting generalized cardinal numbers are convenient tools for describing the powers of vaguely defined objects. In the second part of the paper, basic operations on the generalized cardinals are defined and carefully investigated. Similarities and anomalies in comparison with the classical arithmetic of the usual cardinals are indicated.