Independence results for weak systems of intuitionistic arithmetic

Abstract

AbstractThis paper proves some independence results for weak fragments of Heyting arithmetic by using Kripke models. We present a necessary condition for linear Kripke models of arithmetical theories which are closed under the negative translation and use it to show that the union of the worlds in any linear Kripke model of HA satisfies PA. We construct a two‐node PA‐normal Kripke structure which does not force iΣ2. We prove i∀1⊬ i∃1, i∃1⊬ i∀1, iΠ2⊬ iΣ2and iΣ2⊬ iΠ2. We use Smorynski's operation Σ′ to show HA ⊬ lΠ1.