On Interpretations of Bounded Arithmetic and Bounded Set Theory

In [5], 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 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 is bi-interpretable with the theory of bounded arithmetic I∆0 + 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. Finally, I build on the work of Enayat, Schmerl, and Visser [2] to show that a particular weaker theory of sets is not bi-interpretable with I∆0 + exp. Primary Subject: 03C62