A System of Axiomatic Set Theory

Abstract

The aim of this chapter is to discuss a system of axiomatic set theory. It is a modification of the axiom system due to von Neumann. It adopts the principal idea of von Neumann that the elimination of the undefined notion of a property that occurs in the original axiom system of Zermelo can be accomplished in a way so as to make the resulting axiom system elementary, in the sense of being formalizable in the logical calculus of first order, which contains no other bound variables than individual variables and no accessory rule of inference. The purpose of modifying the von Neumann system is to remain nearer to the structure of the original Zermelo system and at the same time, utilize some of the set-theoretic concepts of the Schroder Logic and of Principia Mathematica that have become familiar to logicians. A considerable simplification results from this arrangement. The theory is not set up as a pure formalism but rather in the usual manner of elementary axiom theory, where one has to deal with propositions that are understood to have a meaning and the reference to the domain of facts to be axiomatized is suggested by the names for the kinds of individuals and for the fundamental predicates.