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  • Cited by 1
  • Print publication year: 2011
  • Online publication date: October 2011

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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