On interpretations of bounded arithmetic and bounded set theory

Notre Dame Journal of Formal Logic 50 (2):141-152 (2009)

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 1 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 IDelta0 + 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.

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Richard Pettigrew
Bristol University

References found in this work

Existence and Feasibility in Arithmetic.Rohit Parikh - 1971 - Journal of Symbolic Logic 36 (3):494-508.
On Interpretations of Arithmetic and Set Theory.Richard Kaye & Tin Lok Wong - 2007 - Notre Dame Journal of Formal Logic 48 (4):497-510.
Die Widerspruchsfreiheit der Allgemeinen Mengenlehre.Wilhelm Ackerman - 1937 - Journal of Symbolic Logic 2 (4):167-167.

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Citations of this work

Constructive Ackermann's Interpretation.Hanul Jeon - 2022 - Annals of Pure and Applied Logic 173 (5):103086.

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