On Interpretations of Arithmetic and Set Theory

Notre Dame Journal of Formal Logic 48 (4):497-510 (2007)
Abstract
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
Keywords Peano arithmetic   finite set theory   interpretations
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DOI 10.1305/ndjfl/1193667707
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Equivalences for Truth Predicates.Carlo Nicolai - 2017 - Review of Symbolic Logic 10 (2):322-356.
The Strength of Extensionality II—Weak Weak Set Theories Without Infinity.Kentaro Sato - 2011 - Annals of Pure and Applied Logic 162 (8):579-646.
Substandard Models of Finite Set Theory.Laurence Kirby - 2010 - Mathematical Logic Quarterly 56 (6):631-642.
On the Concept of Finitism.Luca Incurvati - 2015 - Synthese 192 (8):2413-2436.

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