A note on parameter free Π 1 ‐induction and restricted exponentiation

We characterize the sets of all Π 2 and all \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {B}(\Sigma _{1})$\end{document} (= Boolean combinations of Σ 1 ) theorems of I Π − 1 in terms of restricted exponentiation, and use these characterizations to prove that both sets are not deductively equivalent. We also discuss how these results generalize to n > 0. As an application, we prove that a conservation theorem of Beklemishev stating that I Π − n + 1 is conservative over I Σ − n with respect to \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {B}(\Sigma _{n+1})$\end{document} sentences cannot be extended to Π n + 2 sentences. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim