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 $I{\Delta }_0+\exp$. 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.