Modified Realizability Toposes and Strong Normalization Proofs

This paper is motivated by the discovery that an appropriate quotient SN 3 of the strongly normalising untyped 3-terms (where 3 is just a formal constant) forms a partial applicative structure with the inherent application operation. The quotient structure satises all but one of the axioms of a partial combinatory algebra (pca). We call such partial applicative structures conditionally partial combinatory algebras (c-pca). Remarkably, an arbitrary right-absorptive c-pca gives rise to a tripos provided the underlying intuitionistic predicate logic is given an interpretation in the style of Kreisel's modied realizability, as opposed to the standard Kleene-style realizability. Starting from an arbitrary right-absorptive c-pca U , the tripos-to-topos construction due to Hyland et al. can then be carried out to build a modied realizability topos TOP m (U) of non-standard sets equipped with an equality predicate. Church's Thesis is internally valid in TOP m (K 1) (where the pca K 1 is \Kleene's rst model" of natural numbers) but not Markov's Principle. There is a topos inclusion of SET | the \classical" topos of sets | into TOP m (U); the image of the inclusion is just sheaves for the ::-topology. Separated objects of the ::-topology are characterized. We identify the appropriate notion of pers (partial equivalence relations) in the modied realizability setting and state its completeness properties. The topos TOP m (U) has enough completeness property to provide a category-theoretic semantics for a family of higher type theories which include Girard's System F and the Calculus of Constructions due to Coquand and Huet. As an important application, by interpreting type theories in the topos TOP m (SN 3), a clean semantic explanation of the Tait-Girard style strong normalization argument is obtained. We illustrate how a strong normalization proof for an impredicative and dependent type theory may be assembled from two general \stripping argu-ments" in the framework of the topos TOP m (SN 3). This opens up the possibility of a \generic" strong normalization argument for an interesting class of type theories.