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.
"On Interpretations of Bounded Arithmetic and Bounded Set Theory." Notre Dame J. Formal Logic 50 (2) 141 - 151, 2009. https://doi.org/10.1215/00294527-2009-003