On Interpretations of Bounded Arithmetic and Bounded Set Theory

On Interpretations of Bounded Arithmetic and Bounded Set Theory

Richard Pettigrew

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

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 . 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.

Primary Subjects: 03C62

Keywords: I Delta 0 + exp; finite set theory; interpretations

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1242067707
Digital Object Identifier: doi:10.1215/00294527-2009-003
Mathematical Reviews number (MathSciNet): MR2535581

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References

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Zentralblatt MATH: 0016.19501

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Digital Object Identifier: doi:10.1305/ndjfl/1193667707

Project Euclid: euclid.ndjfl/1193667707

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JSTOR: links.jstor.org

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