ω-models of finite set theory

Abstract

Abstract. 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 this paper we initiate the metamathematical investigation of ω-models of ZF fin . In particular, we present a perspicuous method for constructing recursive nonstandard ω-model of ZF fin without the use of permutations. We then use this method to establish the following central theorem. Theorem A. For every simple graph (A, F ), where F is a set of unordered pairs of A, there is an ω-model M of ZF fin whose universe contains A and which satisfies the following two conditions:(1) There is parameter-free formula ϕ(x, y) such that for all elements a andTheorem A enables us to build a variety of ω-models with special features, in particular:

Every Countable Model of Set Theory Embeds Into Its Own Constructible Universe

“…Specifically, [KW07] shows that the models of PA give rise to all the models of the finite set theory ZFC ¬∞ = ZFC − Inf + ¬Inf + TC, where TC is the assertion that every set has a transitive closure, and conversely the natural numbers of any such model satisfies PA, in inverse fashion, making for a bi-interpretation of these two theories. The curious anomaly here is that while the axiom TC is provable and may be omitted if one replaces the usual foundation axiom in ZFC with the axiom scheme of ∈induction, nevertheless results in [ESV11] show, quite interestingly, that TC is not provable if one uses only the usual foundation axiom. The theme of this article begins in earnest with the following remarkable theorem of Ressayre's.…”

Variations on a Visserian Theme

“…Theorem. (E-Schmerl-Visser [ESV,Theorem 5.1]) ZF fin and PA are not bi-interpretable; indeed ZF fin is not even a sentential retract 5 of ZF fin + TC.…”

Conservativity of transitive closure over weak constructive operational set theory

“…The (quite surprising) role of transitive closure for determining the relation between classical set theory without infinity and Peano Arithmetic has been investigated in [20]; see also [21,12]. Models of Zermelo set theory (with foundation) in which Transitive Closure fails have been given for example in [5,6,22].…”