The theory of classes A modification of von Neumann's system

1. The theory of classes presented in this paper is a simplification of that presented by J. von Neumann in his paper Die Axiomatisierung der Mengenlehre.' However, this paper is written so that it can be read independently of von Neumann's. The principal modifications of his system are the following. (1) The idea of ordered pair is defined in terms of the other primitive concepts of the system. (See Axiom 4.3 below.) (2) A much simpler proof of the well-ordering theorem, based on von Neumann's equivalence axiom (Axiom 2.2 below) is given. (More exactly, the theory of ordinal numbers, on which the proof of the well-ordering theorem is based, is simplified-see ?8.) (3) Functions are not assumed to be defined for all arguments. In place of "A" we have "is undefined." This makes possible a constructive interpretation of the system. (See ?2.)