Bounded finite set theory

We define an axiom schema I Δ 0 S for finite set theory with bounded induction on sets, analogous to the theory of bounded arithmetic, I Δ 0 , and use some of its basic model theory to establish some independence results for various axioms of set theory over I Δ 0 S . Then we ask: given a model M of I Δ 0 , is there a model of I Δ 0 S whose ordinal arithmetic is isomorphic to M ? We show that the answer is yes if M ⊧ Exp .