ω-models of finite set theory [chapter]

Ali Enayat, James H. Schmerl, Albert Visser, Juliette Kennedy, Roman Kossak
Set Theory, Arithmetic, and Foundations of Mathematics  
Finite set theory, here denoted ZF fin , 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 ZF fin 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 ZF fin that is nonstandard (i.e., not isomorphic to the hereditarily finite sets V ω ). In
more » ... paper we initiate the metamathematical investigation of ω-models of ZF fin . In particular, we present a new method for building ω-models of ZF fin that leads to a perspicuous construction of recursive nonstandard ω-models of ZF fin without the use of permutations. Furthermore, we show that every recursive model of ZF fin is an ω-model. The central theorem of the paper is the following:
doi:10.1017/cbo9780511910616.004 fatcat:gc42okd7s5dwtgbf7adkdkfmrq