The Model Companion of ZF

Abstract

We prove that the theory ZF has a model companion and we describe an axiom system for it. The notion of a model companion was introduced by E. Bers as a generalization of the notion of a model completion [3, ?5]. In this paper we prove that ZF has a model companion and describe a set of axioms for it. This model companion, however, resembles more a theory of order (Theorem 3) than a set theory, and therefore, while supplying an interesting example for model theory it does not shed any new light on set theory. We feel that this example demonstrates that by generalizing to model companions one may lose the interesting relations between a theory and its model completion. We deal with a language with a single binary relation e. In a given model we say that there is an e-chain leading from b to c if for some a a (O < n) b E a1 E *.. E a E c. A closed e-chain is a chain leading from some element to itself. First we define the theory S: s \V1.. xn(x 1l x2V ...* V x n_ /xn \/ Xn /XdlI< n < wl. S claims that there are no closed e-chains. It is a universal theory (all the axioms are universal sentences) and S C ZF by the axiom of regularity. We want to show that S is the universal part of ZF. We show more: Theorem 1. If ilr is a universal sentence then either S F i/ or ZF F 7 ,lr. Proof. We assume that not S F ilr and show that ZF W]. Let Vb be Vx .. x.xnj() where ? is quantifier free. We can assume that 0b is the disjunction of atomic formulas or their negations: it is at least a conjunction ?01A ... A O. of such formulas and if not S F V To then for some i < k not S F Vxi(T). If we have shown that ZF F 7 Vx?0. then ZF F _i Vxi0. Thus we assume that ?b(T) = F1 (x ) V * V Fk(x) where every F is of Received by the editors March 29, 1974. AMS (MOS) subject classifications (1970). Primary 02H05.