Induction rules in bounded arithmetic

  • Emil JeřábekEmail author


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.


Bounded arithmetic Parameter-free induction Induction rule Partial conservativity Reflection principle 

Mathematics Subject Classification

03F30 03F20 



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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Authors and Affiliations

  1. 1.Institute of MathematicsThe Czech Academy of SciencesPrague 1Czech Republic

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