Notre Dame Journal of Formal Logic

On Interpretations of Bounded Arithmetic and Bounded Set Theory

Richard Pettigrew

Abstract

In "On interpretations of arithmetic and set theory," Kaye and Wong proved the following result, which they considered to belong to the folklore of mathematical logic.

Theorem The first-order theories of Peano arithmetic and Zermelo-Fraenkel set theory with the axiom of infinity negated are bi-interpretable.

In this note, I describe a theory of sets that is bi-interpretable with the theory of bounded arithmetic I Δ0 +exp . Because of the weakness of this theory of sets, I cannot straightforwardly adapt Kaye and Wong's interpretation of the arithmetic in the set theory. Instead, I am forced to produce a different interpretation.

Article information

Source
Notre Dame J. Formal Logic, Volume 50, Number 2 (2009), 141-151.

Dates
First available in Project Euclid: 11 May 2009

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1242067707

Digital Object Identifier
doi:10.1215/00294527-2009-003

Mathematical Reviews number (MathSciNet)
MR2535581

Zentralblatt MATH identifier
1183.03029

Subjects
Primary: 03C62: Models of arithmetic and set theory [See also 03Hxx]

Keywords
I Delta 0 + exp finite set theory interpretations

Citation

Pettigrew, Richard. On Interpretations of Bounded Arithmetic and Bounded Set Theory. Notre Dame J. Formal Logic 50 (2009), no. 2, 141--151. doi:10.1215/00294527-2009-003. https://projecteuclid.org/euclid.ndjfl/1242067707


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References

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