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December 2013 On the optimality of conservation results for local reflection in arithmetic
A. Cordón-Franco, A. Fernández-Margarit, F. F. Lara-Martín
J. Symbolic Logic 78(4): 1025-1035 (December 2013). DOI: 10.2178/jsl.7804010

Abstract

Let $T$ be a recursively enumerable theory extending Elementary Arithmetic $\rm{EA}$. L. D. Beklemishev proved that the $\Sigma_2$ local reflection principle for $T$, $\mathsf{Rfn}_{\Sigma_2}(T)$, is conservative over the $\Sigma_1$ local reflection principle, $\mathsf{Rfn}_{\Sigma_1}(T)$, with respect to boolean combinations of $\Sigma_1$-sentences; and asked whether this result is best possible. In this work we answer Beklemishev's question by showing that $\Pi_2$-sentences are not conserved for $T = \rm{EA}{}+{}\textit{"f is total"}$, where $f$ is any nondecreasing computable function with elementary graph. We also discuss how this result generalizes to $n > 0$ and obtain as an application that for $n > 0$, $I\Pi_{n+1}^-$ is conservative over $I\Sigma_n$ with respect to $\Pi_{n+2}$-sentences.

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A. Cordón-Franco. A. Fernández-Margarit. F. F. Lara-Martín. "On the optimality of conservation results for local reflection in arithmetic." J. Symbolic Logic 78 (4) 1025 - 1035, December 2013. https://doi.org/10.2178/jsl.7804010

Information

Published: December 2013
First available in Project Euclid: 5 January 2014

zbMATH: 1316.03036
MathSciNet: MR3156510
Digital Object Identifier: 10.2178/jsl.7804010

Rights: Copyright © 2013 Association for Symbolic Logic

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Vol.78 • No. 4 • December 2013
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