Variations on Realizability: Realizing the Propositional Axiom of Choice

Early investigators of realizability were interested in metamathematical questions. In keeping with the traditions of the time they concentrated on interpretations of one formal system in another. They considered an ad hoc collection of increasingly ingenious interpretations to establish consistency, independence and conservativity results. van Oosten’s contribution to the Workshop (see van Oosten [46]) gave inter alia an account of these concerns from a modern perspective. In the early days of categorical logic one considered realizability as providing models for constructive mathematics; while the metamathematics could be retrieved by ‘coding’ the models, that aspect took a back seat. In the first instance realizability provided toposes, that is models for constructive type theory; but it also can be used to model stronger systems of impredicative constructive set theory. In time it was recognized that the mathematical structures arising from realizability provided models for more exotic non-classical theories of interest. Work then focused in particular on models for impredicative polymorphic calculi such as System F (Girard [12]) and the Calculus of Constructions (Coquand and Huet [7]), and on Synthetic Domain Theory (Hyland [19] and Taylor [41]). The use of realizability in this context has a quite different character from its earlier metamathematical use. For details of realizability models impredicative type theories the reader may consult Crole [8]. For formal expositions of Synthetic Domain Theory informed by the realizability experience see Reus [34] and Reus and Streicher [35]. The focus of this paper is on the axiom of choice