4,579 research outputs found
Provably Total Functions of Arithmetic with Basic Terms
A new characterization of provably recursive functions of first-order
arithmetic is described. Its main feature is using only terms consisting of 0,
the successor S and variables in the quantifier rules, namely, universal
elimination and existential introduction.Comment: In Proceedings DICE 2011, arXiv:1201.034
On axiom schemes for T-provably Δ1 formulas
This paper investigates the status of the fragments of Peano Arithmetic obtained by restricting induction, collection and least number axiom schemes to formulas which are Δ1 provably in an arithmetic theory T. In particular, we determine the provably total computable functions of this kind of theories. As an application, we obtain a reduction of the problem whether IΔ0+¬exp implies BΣ1 to a purely recursion-theoretic question.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
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
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