A cardinality theory for vaguely defined objects-problems of inequalities and applications

In this paper we show some unconventional techniques of applying mathematics. More precisely, making use of the Łukasiewicz logic we build a nonclassical cardinality theory for vaguely defined objects which are more general constructions than sets. Our attention is focused on questions related to inequalities and comparisons. They seem to be essential from the applicational viewpoint. Resulting generalized cardinal numbers, counterparts of usual cardinals occurring in the classical theory, are convenient tools for quantitative description and analysis of the objects, the information about which is imperfect, i.e. is vague, imprecise, incomplete, etc. What enhanced the motivation for this research was the area of possible applications which comprises information and computer sciences, approximate reasoning, decision and control theories, nonclassical branches of mathematics like, for instance, fuzzy topological spaces, mathematical social sciences, engineering, etc.