Notre Dame Journal of Formal Logic

On Interpretations of Arithmetic and Set Theory

Richard Kaye and Tin Lok Wong

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.

Article information

Source
Notre Dame J. Formal Logic, Volume 48, Number 4 (2007), 497-510.

Dates
First available in Project Euclid: 29 October 2007

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1193667707

Digital Object Identifier
doi:10.1305/ndjfl/1193667707

Mathematical Reviews number (MathSciNet)
MR2357524

Zentralblatt MATH identifier
1137.03019

Subjects
Primary: 03H15: Nonstandard models of arithmetic [See also 11U10, 12L15, 13L05]
Secondary: 03C62: Models of arithmetic and set theory [See also 03Hxx]

Keywords
Peano arithmetic finite set theory interpretations

Citation

Kaye, Richard; Wong, Tin Lok. On Interpretations of Arithmetic and Set Theory. Notre Dame J. Formal Logic 48 (2007), no. 4, 497--510. doi:10.1305/ndjfl/1193667707. https://projecteuclid.org/euclid.ndjfl/1193667707


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