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  1. Fragments of Arithmetic and True Sentences.Andrés Cordón-Franco, Alejandro Fernández-Margarit & F. Félix Lara-Martín - 2005 - Mathematical Logic Quarterly 51 (3):313-328.
    By a theorem of R. Kaye, J. Paris and C. Dimitracopoulos, the class of the Πn+1-sentences true in the standard model is the only consistent Πn+1-theory which extends the scheme of induction for parameter free Πn+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 (...)
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  • A Note on Parameter Free Π1‐Induction and Restricted Exponentiation.A. Cordón‐Franco, A. Fernández‐Margarit & F. F. Lara‐Martín - 2011 - Mathematical Logic Quarterly 57 (5):444-455.
    We characterize the sets of all Π2 and all equation image theorems of IΠ−1 in terms of restricted exponentiation, and use these characterizations to prove that both sets are not deductively equivalent. We also discuss how these results generalize to n > 0. As an application, we prove that a conservation theorem of Beklemishev stating that IΠ−n + 1 is conservative over IΣ−n with respect to equation image sentences cannot be extended to Πn + 2 sentences. © 2011 WILEY-VCH Verlag (...)
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  • On Axiom Schemes for T-Provably $${\Delta_{1}}$$ Δ 1 Formulas.A. Cordón-Franco, A. Fernández-Margarit & F. F. Lara-Martín - 2014 - Archive for Mathematical Logic 53 (3-4):327-349.
  • Local Induction and Provably Total Computable Functions.Andrés Cordón-Franco & F. Félix Lara-Martín - 2014 - Annals of Pure and Applied Logic 165 (9):1429-1444.
    Let Iπ2 denote the fragment of Peano Arithmetic obtained by restricting the induction scheme to parameter free Π2Π2 formulas. Answering a question of R. Kaye, L. Beklemishev showed that the provably total computable functions of Iπ2 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 Iπ2. In this way, we obtain a more direct answer to Kaye's question, avoiding the (...)
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