Intuitionistic validity in T-normal Kripke structures

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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.

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Supported in part by NSF Grant DMS-8902480.