Journal of Symbolic Logic

On the optimality of conservation results for local reflection in arithmetic

A. Cordón-Franco, A. Fernández-Margarit, and F. F. Lara-Martín

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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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J. Symbolic Logic, Volume 78, Issue 4 (2013), 1025-1035.

First available in Project Euclid: 5 January 2014

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Cordón-Franco, A.; Fernández-Margarit, A.; Lara-Martín, F. F. On the optimality of conservation results for local reflection in arithmetic. J. Symbolic Logic 78 (2013), no. 4, 1025--1035. doi:10.2178/jsl.7804010.

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