Axiomatizing higher-order Kleene realizability

Abstract

Kleene's realizability interpretation (1945) for first-order arithmetic was shown by Hyland (1982) to fit into the internal logic of an elementary topos, the “Effective topos” Eff . In this paper it is shown, that there is an internal realizability definition in Eff , i.e. a syntactical translation of the internal language of Eff into itself of form “ n realizes ϕ” (where n is a variable of type the natural numbers object of Eff ), which extends Kleene's definition, and such that for sentences ϕ, the equivalence ϕ[harr]∃ n( n realizes ϕ) is true in Eff . The internal realizability definition depends on finding separated covers for non-separated objects of Eff . However, for the objects arising in higher-order arithmetic, canonical covers are available, which are definable in higher-order arithmetic. These canonical covers yield “covering axioms”. It is shown that these covering axioms, together with uniformity principles and Extended Church's Thesis, axiomatize a formalized extension of Kleene realizability to a constructive system of higher-order arithmetic. The details are worked out for second and third-order arithmetic, but the method can be extended to any order. As an application, it is shown in the final section that a certain completeness property of the (internal) category of “modest sets” can be derived in third-order arithmetic from the realizability axioms.