On Models of Zermelo—Fraenkel Set Theory Satisfying the Axiom of Gonstructibility

Abstract

This chapter illustrates that there exists a minimal model of the Zermelo–Fraenkel set theory and outlines an alternative proof of this model and adds some remarks concerning sets that are definable in models of set theory with the axiom of constructivity. The proofs shall be based on the axioms of Zermelo-Fraenkel (with the axiom of regularity but without the axiom of choice) and on an additional axiom that states the existence of a set whose cardinal number is a strongly inaccessible aleph. It is assumed that the notion of a Godel number of a formula has been defined in set theory and also that the arithmetical counterparts of the usual metamathematical notions have also been defined. The chapter identifies a formula with the corresponding Godel number. The set of the (Godel numbers of the) axioms of Zermelo-Fraenkel is denoted by Ax .