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-model of ZFfin without the use of permutations. Furthermore,
we show that every recursive model of ZFfin is an ω-model
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