Open Access
2007 On Interpretations of Arithmetic and Set Theory
Richard Kaye, Tin Lok Wong
Notre Dame J. Formal Logic 48(4): 497-510 (2007). DOI: 10.1305/ndjfl/1193667707


This paper starts by investigating Ackermann's interpretation of finite set theory in the natural numbers. We give a formal version of this interpretation from Peano arithmetic (PA) to Zermelo-Fraenkel set theory with the infinity axiom negated (ZF−inf) and provide an inverse interpretation going the other way. In particular, we emphasize the precise axiomatization of our set theory that is required and point out the necessity of the axiom of transitive containment or (equivalently) the axiom scheme of ∈-induction. This clarifies the nature of the equivalence of PA and ZF−inf and corrects some errors in the literature. We also survey the restrictions of the Ackermann interpretation and its inverse to subsystems of PA and ZF−inf, where full induction, replacement, or separation is not assumed. The paper concludes with a discussion on the problems one faces when the totality of exponentiation fails, or when the existence of unordered pairs or power sets is not guaranteed.


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Richard Kaye. Tin Lok Wong. "On Interpretations of Arithmetic and Set Theory." Notre Dame J. Formal Logic 48 (4) 497 - 510, 2007.


Published: 2007
First available in Project Euclid: 29 October 2007

zbMATH: 1137.03019
MathSciNet: MR2357524
Digital Object Identifier: 10.1305/ndjfl/1193667707

Primary: 03H15
Secondary: 03C62

Keywords: finite set theory , interpretations , Peano Arithmetic

Rights: Copyright © 2007 University of Notre Dame

Vol.48 • No. 4 • 2007
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