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# ω-models of finite set theory

## Summary

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

Theorem A. For every graph (A, F), where F is a set of unordered pairs of A, there is an ω-model m of ZFfin whose universe contains A and which satisfies the following conditions:

(1) (A, F) is definable in m;

(2) Every element of m is definable in (m, a)aA;

(3) If (A, F) is pointwise definable, then so is m;

(4) Aut(m) ≅ Aut(A, F).

Theorem A enables us to build a variety of ω-models with special features, in particular:

Corollary 1. Every group can be realized as the automorphism group of an ω-model of ZFfin.

Corollary 2. For each infinite cardinal κ there are 2κrigid nonisomorphic ω-models of ZFfinof cardinality κ. […]

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