Model of ZF Set Theory

The goal of this article is to construct a language of the ZF set theory and to develop a notational and conceptual base which facilitates a convenient usage of the language. and 2] provide the notation and terminology for this paper. For simplicity we adopt the following rules: k, n denote natural numbers, D denotes a non empty set, a is arbitrary, and p, q denote nite sequences of elements of N. The non empty subset VAR of N is deened by: (Def. 1) VAR = fk : 5 kg: We now state the proposition (1) VAR = fk : 5 kg: A variable is an element of VAR. Let us consider n. The functor x n yielding a variable is deened as follows: (Def. 2) x n = 5 + n: One can prove the following proposition (3) 2 x n = 5 + n: In the sequel x, y, z, t are variables. Let us consider x. Then hxi is a nite sequence of elements of N. Let us consider x, y. The functor x=y yields a nite sequence of elements of N and is deened as follows: (Def. 3) x=y = h0i a hxi a hyi: The functor xxy yielding a nite sequence of elements of N is deened by: (Def. 4) xxy = h1i a hxi a hyi: The following propositions are true: (4) x=y = h0i a hxi a hyi: (5) xxy = h1i a hxi a hyi: (6) If x=y = z=t; then x = z and y = t: (7) If xxy = zt; then x = z and y = t: Let us consider p. The functor :p yields a nite sequence of elements of N and is deened by: (Def. 5) :p = h2i a p: 1 Supported by RPBP.III-24.C1. 2 The proposition (2) has been removed.