A Buchholz derivation system for the ordinal analysis of KP+Π₃-reflection

Abstract

In this paper we introduce a notation system for the infinitary derivations occurring in the ordinal analysis of KP+Π_3-Reflection due to Michael Rathjen. This allows a finitary ordinal analysis of KP+Π₃-Reflection. The method used is an extension of techniques developed by Wilfried Buchholz, namely operator controlled notation systems for RS^∞-derivations. Similarly to Buchholz we obtain a characterisation of the provably recursive functions of KP+Π₃-Reflection as <-recursive functions where < is the ordering on Rathjen’s ordinal notation system 𝒯(K). Further we show a conservation result for Π⁰₂-sentences.