On Bounded ∑ 1 1 Polynomial Induction

We characterize the bounded first order consequences of theory in U 2 1 terms of a limited use of exponentiation, we construct a simulation of U 2 1 by the quantified propositional calculus, and we prove that U 2 1 is not conservative over IΔ0 and that it is stronger than a conservative Δ 1 1,b -extension of S 2. As corollaries we obtain that U 2 1 is not conservative over TNC and that ∑ j b -consequences of U 1 2 are finitely axiomatizable (j ≥ 2). We also show that Ů 2 1 plus a version of ∏ 1 1,b -SEP is conservative over U 2 1 (BD) w.r.t. bounded formulas.