On interpretations of bounded arithmetic and bounded set theory

TL;DR

This paper explores the interpretability relations between bounded arithmetic and a corresponding set theory, extending known results about classical theories to weaker, bounded theories.

Contribution

It introduces a set theory that is bi-interpretable with bounded arithmetic IDelta0 + exp, adapting the interpretation approach to weaker theories.

Findings

Established bi-interpretability between bounded arithmetic and a new set theory

Demonstrated limitations of direct interpretation due to theory weakness

Provided a novel interpretation method for bounded theories

Abstract

In a recent paper, 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 ZF with the axiom of infinity negated are bi-interpretable: that is, they are mutually interpretable with interpretations that are inverse to each other. In this note, I describe a theory of sets that stands in the same relation to the bounded arithmetic IDelta0 + exp. Because of the weakness of this theory of sets, I cannot straightforwardly adapt Kaye and Wong's interpretation of arithmetic in set theory. Instead, I am forced to produce a different interpretation.