Local Induction and Provably Total Computable Functions: A Case Study

Let IΠ−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 IΠ−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 IΠ−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.Ministerio de Ciencia e Innovación MTM2008–0643

Local induction and provably total computable functions

Let I¦− 2 denote the fragment of Peano Arithmetic obtained by restricting the induction scheme to parameter free ¦2 formulas. Answering a question of R. Kaye, L. Beklemishev showed that the provably total computable functions of I¦− 2 are, precisely, the primitive recursive ones. In this work we give a new proof of this fact through an analysis of certain local variants of induction principles closely related to I¦− 2 . 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. Our methods are model–theoretic and allow for a general study of I¦− n+1 for all n ¸ 0. In particular, we derive a new conservation result for these theories, namely that I¦− n+1 is ¦n+2–conservative over I§n for each n ¸ 1.Ministerio de Ciencia e Innovación MTM2008–06435Ministerio de Ciencia e Innovación MTM2011–2684

Fragments of Arithmetic and true sentences

By a theorem of R. Kaye, J. Paris and C. Dimitracopoulos, the class of the ¦n+1–sentences true in the standard model is the only (up to deductive equivalence) consistent ¦n+1–theory which extends the scheme of induction for parameter free ¦n+1–formulas. Motivated by this result, we present a systematic study of extensions of bounded quantifier complexity of fragments of first–order Peano Arithmetic. Here, we improve that result and show that this property describes a general phenomenon valid for parameter free schemes. As a consequence, we obtain results on the quantifier complexity, (non)finite axiomatizability and relative strength of schemes for ¢n+1–formulas.Junta de Andalucía TIC-13