Annals of Pure and Applied Logic 136 (1-2):56-74 (2005)

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Abstract
We study the arithmetical schema asserting that every eventually decreasing elementary recursive function has a limit. Some other related principles are also formulated. We establish their relationship with restricted parameter-free induction schemata. We also prove that the same principle, formulated as an inference rule, provides an axiomatization of the Σ2-consequences of IΣ1.Using these results we show that ILM is the logic of Π1-conservativity of any reasonable extension of parameter-free Π1-induction schema. This result, however, cannot be much improved: by adapting a theorem of D. Zambella and G. Mints we show that the logic of Π1-conservativity of primitive recursive arithmetic properly extends ILM.In the third part of the paper we give an ordinal classification of -consequences of the standard fragments of Peano arithmetic in terms of reflection principles. This is interesting in view of the general program of ordinal analysis of theories, which in the most standard cases classifies Π-classes of sentences
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DOI 10.1016/j.apal.2005.05.005
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References found in this work BETA

Systems of Logic Based on Ordinals.Alan Mathison Turing - 1939 - London: Printed by C.F. Hodgson & Son.
Proof-Theoretic Analysis by Iterated Reflection.Lev D. Beklemishev - 2003 - Archive for Mathematical Logic 42 (6):515-552.
Transfinite Recursive Progressions of Axiomatic Theories.Solomon Feferman - 1962 - Journal of Symbolic Logic 27 (3):259-316.
The Interpretability Logic of Peano Arithmetic.Alessandro Berarducci - 1990 - Journal of Symbolic Logic 55 (3):1059-1089.
Iterated Local Reflection Versus Iterated Consistency.Lev Beklemishev - 1995 - Annals of Pure and Applied Logic 75 (1-2):25-48.

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Citations of this work BETA

Axiomatization of Provable N-Provability.Evgeny Kolmakov & Lev Beklemishev - 2019 - Journal of Symbolic Logic 84 (2):849-869.
Transductions in Arithmetic.Albert Visser - 2016 - Annals of Pure and Applied Logic 167 (3):211-234.
Marginalia on a Theorem of Woodin.Rasmus Blanck & Ali Enayat - 2017 - Journal of Symbolic Logic 82 (1):359-374.

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