The foundations of arithmetic in finite bounded Zermelo set theory

Cahiers du Centre de Logique 17:99-118 (2010)
Abstract
In this paper, I pursue such a logical foundation for arithmetic in a variant of Zermelo set theory that has axioms of subset separation only for quantifier-free formulae, and according to which all sets are Dedekind finite. In section 2, I describe this variant theory, which I call ZFin0. And in section 3, I sketch foundations for arithmetic in ZFin0 and prove that certain foundational propositions that are theorems of the standard Zermelian foundation for arithmetic are independent of ZFin0.<br><br>An equivalent theory of sets and an equivalent foundation for arithmetic was introduced by John Mayberry and developed by the current author in his doctoral thesis. In that thesis, the independence results mentioned above are proved using proof-theoretic methods. In this paper, I offer model-theoretic proofs of the central independence results using the technique of cumulation models, which was introduced by Steve Popham, a doctoral student of Mayberry<br>from the early 1980s.
Keywords set theory  arithmetic
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