A note on parameter free Π1-induction and restricted exponentiation
Andrés Cordón–Franco, A. Fernández‐Margarit, Francisco Martín · Mathematical logic quarterly · 2011
We characterize the sets of all Π2 and all \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {B}(\Sigma _{1})$\end{document} (= Boolean combinations of Σ1) theorems of IΠ−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 IΠ−n + 1 is conservative over IΣ−n with respect to \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {B}(\Sigma _{n+1})$\end{document} sentences cannot be extended to Πn + 2 sentences. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim