Intuitionistic Open Induction and Least Number Principle and the Buss Operator

Notre Dame Journal of Formal Logic 39 (2):212-220 (1998)
Abstract
In "Intuitionistic validity in -normal Kripke structures," Buss asked whether every intuitionistic theory is, for some classical theory , that of all -normal Kripke structures for which he gave an r.e. axiomatization. In the language of arithmetic and denote PA plus Open Induction or Open LNP, and are their intuitionistic deductive closures. We show is recursively axiomatizable and , while . If proves PEM but not totality of a classically provably total Diophantine function of , then and so . A result due to Wehmeier then implies . We prove is not -conservative over . If , then is not closed under MR or Friedman's translation, so range (). Both and are closed under the negative translation
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Sarah Buss (1999). Practical Induction. Philosophical Review 108 (4):571-575.
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