Universals as Generalized Sets

Abstract

This work is an attenpt to solve the problem of finding a natural generalization of set theory. The kind of generalization we would like to have is illustrated in the way that set theory generalizes number theory. Not only are the natural nuribers identified with certain sets, but the totality of alL natural numbers is itself identified with some object in the universe of sets. The existence of the set ur of aII natural numbers is postulated in the axiom of infinity of ZFC. So, we would like a theory of some domain of "generalized sets" such that each set can be identified with some object in this domain, and such that the totality of all sets can be construed as some particular generalized set.