Local Induction and Provably Total Computable Functions: A Case Study

Abstract

Let $I\Pi_2^-$ denote the fragment of Peano Arithmetic obtained by restricting the induction scheme to parameter free Π₂ formulas. Answering a question of R. Kaye, L. Beklemishev showed that the provably total computable functions (p.t.c.f.) of $I\Pi_2^-$ are, precisely, the primitive recursive ones. In this work we give a new proof of this fact through an analysis of the p.t.c.f. of certain local versions of induction principles closely related to $I\Pi_2^-$. This analysis is essentially based on the equivalence between local induction rules and restricted forms of iteration. In this way, we obtain a more direct answer to Kaye’s question, avoiding the metamathematical machinery (reflection principles, provability logic,...) needed for Beklemishev’s original proof.