OnModel Theory for Intuitionistic BoundedArithmetic withApplications to IndependenceResults

IPV is IPV (which is essentially IS 2 ) with polynomial-induction on Σ 1 -formulas disjoined with arbitrary formulas in which the induction variable does not occur. This paper proves that IPV is sound and complete with respect to Kripke structures in which every world is a model of CPV (essentially S 2 ). Thus IPV is sound with respect to such structures. In this setting, this is a strengthening of the usual completeness and soundness theorems for first-order intuitionistic theories. Using Kripke structures a conservation result is proved for PV1 over IPV . Cook-Urquhart and Kraj́ıček-Pudlák have proved independence results stating that it is consistent with IPV and PV that extended Frege systems are super. As an application of Kripke models for IPV , we give a proof of a strengthening of Cook and Urquhart’s theorem using the model-theoretic construction of Kraj́ıček and Pudlák.