On the Optimality of Conservation Results for Local Reflection in Arithmetic
Andrés Cordón–Franco, A. Fernández‐Margarit, Francisco Martín · Journal of Symbolic Logic · 2013
Abstract LetTbe a recursively enumerable theory extending Elementary Arithmetic EA. L. D. Beklemishev proved that the Σ2local reflection principle forT, (T), is conservative over the Σ1local reflection principle, (T), with respect to boolean combinations of Σ1-sentences; and asked whether this result is best possible. In this work we answer Beklemishev's question by showing that Π2-sentences are not conserved forT= EA + “f is total,” wherefis any nondecreasing computable function with elementary graph. We also discuss how this result generalizes ton> 0 and obtain as an application that forn> 0, is conservative overIΣnwith respect to Πn+2-sentences.