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