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Semi-honest subrecursive degrees and the collection rule in arithmetic

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Abstract

By a result of L.D. Beklemishev, the hierarchy of nested applications of the \(\Sigma _1\)-collection rule over any \(\Pi _2\)-axiomatizable base theory extending Elementary Arithmetic collapses to its first level. We prove that this result cannot in general be extended to base theories of arbitrary quantifier complexity. In fact, given any recursively enumerable set of true \(\Pi _2\)-sentences, S, we construct a sound \((\Sigma _2 \! \vee \! \Pi _2)\)-axiomatized theory T extending S such that the hierarchy of nested applications of the \(\Sigma _1\)-collection rule over T is proper. Our construction uses some results on subrecursive degree theory obtained by L. Kristiansen.

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Acknowledgements

This work was partially supported by the Ministerio de Ciencia e Innovación (Spanish Government) under the Grant PID2020-116773GB-I00.

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  1. Dpto. Ciencias de la Computación e Inteligencia Artificial, Universidad de Sevilla, C/ Tarfia s/n, 41012, Sevilla, Spain

    Andrés Cordón-Franco & F. Félix Lara-Martín

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  1. Andrés Cordón-Franco
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  2. F. Félix Lara-Martín
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Correspondence to Andrés Cordón-Franco.

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Cordón-Franco, A., Lara-Martín, F.F. Semi-honest subrecursive degrees and the collection rule in arithmetic. Arch. Math. Logic 63, 163–180 (2024). https://doi.org/10.1007/s00153-023-00889-z

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