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ω-Models of finite set theory

By Ali Enayat, James Schmerl and A. Visser


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

Topics: Wijsbegeerte
Year: 2008
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