On the optimality of conservation results for local reflection in arithmetic

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.