Intuitionistic Validity in T-Normal Kripke Structures

Abstract Let T be a first-order theory. A T-normal Kripke structure is one in which every world is a classical model of T. This paper gives a characterization of the intuitionistic theoryHT of sentences intuitionistically valid (forced) in all T-normal Kripke structures and proves the corresponding soundness and completeness theorems. For Peano arithmetic (PA), the theoryHPA is a proper subtheory of Heyting arithmetic (HA), so HA is complete but not sound for PA-normal Kripke structures.