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Induction rules in bounded arithmetic

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Abstract

We study variants of Buss’s theories of bounded arithmetic axiomatized by induction schemes disallowing the use of parameters, and closely related induction inference rules. We put particular emphasis on \(\hat{\varPi }^{b}_i\) induction schemes, which were so far neglected in the literature. We present inclusions and conservation results between the systems (including a witnessing theorem for \(T^i_2\) and \(S^i_2\) of a new form), results on numbers of instances of the axioms or rules, connections to reflection principles for quantified propositional calculi, and separations between the systems.

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Notes

  1. Warning: the proof of Theorem 27, which effectively claims that \(\hat{\varSigma }^{b}_i\hbox {-} (P)IND^- \equiv \hat{\varSigma }^{b}_i\hbox {-} (P)IND^R \), is incorrect.

  2. Their setup includes modular counting gates, but most of the results work also in the usual setup.

  3. In fact, weaker assumptions suffice: it is enough if \({\mathbb {Q}}\), \(\omega \sqcup 1\), and \(\omega ^*\sqcup 1\) do not embed in P, where \(\sqcup \) denotes disjoint union of posets.

  4. The one possible exception is that we used a couple of times the fact that every bounded sentence is provable or refutable in the base theory. This is not literally true in the relativized setting, but it may be replaced by the weaker property that every bounded sentence is equivalent to a Boolean combination of sentences of the form \(\alpha (k)\) for standard constants k.

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Acknowledgements

I would like to thank Andrés Cordón-Franco and Félix Lara-Martín for stimulating discussions of the topic, and for drawing my attention to [28]. I am grateful to the anonymous reviewer for many helpful suggestions.

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  1. Institute of Mathematics, The Czech Academy of Sciences, Žitná 25, 115 67, Prague 1, Czech Republic

    Emil Jeřábek

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Supported by Center of Excellence CE-ITI under the grant P202/12/G061 of GA ČR. The Institute of Mathematics of the Czech Academy of Sciences is supported by RVO: 67985840.

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Jeřábek, E. Induction rules in bounded arithmetic. Arch. Math. Logic 59, 461–501 (2020). https://doi.org/10.1007/s00153-019-00702-w

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