Generalized Cardinal Numbers

Abstract

The notion of equipotency, introduced in the previous chapter, allows us to state if two VD-objects are ‘equinumerous’, i.e. to state if they are identical with respect to their powers (cardinalities). Instead, the aim of this chapter is to introduce and to investigate basic properties of mathematical tools allowing ones to express the powers of VD-objects, in other words — to describe the amount of elements in a VD-object. We feel that the main difficulty, and, simultaneously, the main difference in comparison with sets and their cardinality theory, lies in the nebular feature of VD-objects. That feature causes that the elements from M are in a VD-object only ‘to a degree’, which is expressed by means of a maybe imprecisely determined real number from I, generally distinct for each x ∈ M. The tools used to express the powers of VD-objects will be called generalized cardinal numbers. Shortly speaking, they are some convex functions CN → I, i.e. some special convex VD-object in CN. In the further chapters, their applications, inequalities and arithmetic will be discussed.