ω-Models of finite set theory

Finite set theory, here denoted ZFfin, is the theory ob- tained by replacing the axiom of infinity by its negation in the usual axiomatization of ZF (Zermelo-Fraenkel set theory). An !-model of ZFfin is a model in which every set has at most finitely many elements (as viewed externally). Mancini and Zambella (2001) em- ployed the Bernays-Rieger method of permutations to construct a recursive !-model of ZFfin that is nonstandard (i.e., not isomor- phic to the hereditarily finite sets V!). In this paper we initiate the metamathematical investigation of !-models of ZFfin. In par- ticular, we present a new method for building !-models of ZFfin that leads to a perspicuous construction of recursive nonstandard !-models of ZFfin without the use of permutations. Furthermore, we show that every recursive model of ZFfin is an !-model. The central theorem of the paper is the following: