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On axiom schemes for T-provably \({\Delta_{1}}\) formulas

Abstract

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 \({\Delta_1}\) 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 \({I\Delta_0 + \neg \mathit{exp}}\) implies \({B\Sigma_1}\) to a purely recursion-theoretic question.

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Correspondence to A. Cordón-Franco.

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Cordón-Franco, A., Fernández-Margarit, A. & Lara-Martín, F.F. On axiom schemes for T-provably \({\Delta_{1}}\) formulas. Arch. Math. Logic 53, 327–349 (2014). https://doi.org/10.1007/s00153-014-0368-9

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Keywords

  • Fragments of Peano Arithmetic
  • \({\Delta_1}\) formulas
  • Provably total computable functions

Mathematics Subject Classification

  • 03F30
  • 03D20