The role of classes in set theory

This chapter discusses various systems of set theory that admit, besides sets, classes. Such systems view the sets from the same point of view as the Zermelo–Fraenkel (ZF) set theory “ZF” or from a similar point of view. The language of ZF contains just one kind of individual variable, for which lowercase letters are used. In a general mathematical language the range of the individual variable, the universe of discourse, consists of objects. It is natural to assume that each of these objects is a member of some set. This is in accordance with one of the tacit principles of Cantor's naive set theory that every object can serve as a building block for sets. That set can be, for example, a set that contains just this single object. However, some of the systems of set theory discussed in the present chapter abandon this principle. Yet, as far as ZF is concerned, this principle remains valid; it is implemented by the axiom of pairing. In the version of ZF presented in the chapter, it is assumed that all objects are sets. This is done for the purpose of convenience. Permitting the existence of elements that are not sets would have necessitated some rather trivial changes throughout the chapter.