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
