Abstract
This chapter discusses properties of realizability and modified realizability interpretations for intuitionistic arithmetic HA and intuitionistic arithmetic in all finite types HA ω . The chapter describes the formal systems and presents the model hereditarily recursive operations (HRO) and hereditarily effective operations (HEO) for the intensional and the extensional version of HA ω respectively. The chapter characterizes the axiomatically formulae that can be proved in HA, resp. HA ω to be realizable, resp. modified realizable, resp. Dialectica interpretable. The chapter uses these results and the models HRO, HEO for conservative extension results and consistency results (e.g., consistency of HA with Markov's schema and Church's thesis, HAW is conservative over HA, consistency of certain “axioms of choice” for HRO, HEO) and proves theoretic-closure conditions.
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