Domain Theory and Realizability over Scott’s D∞ in Constructive Set Theory

The first nontrivial model of the untyped l-calculus was Scott's D∞ model, and it has exerted a great influence on the semantics of l-calculi and the theory of programming languages. If one wants to develop domain theory in a constructive set theory such as CZF, though, one faces the problem that the notion of complete partial order implicitly uses the powerset axiom. However, most of domain theory can be attained with partial orders that are complete only with regard to denumerable ascending chains. In the chapter, it is shown that this development can be carried out on the basis of a fragment of CZF. The existence of various D∞-models, however, is not provable in CZF alone but can be carried out in standard extensions of CZF with an appropriate form of the Regular Extension Axiom (which is still interpretable in Martin- Löf-type theory). Using the denizens of these models as realisers for realisability interpretations of set theory, it is shown that they provide universes of intuitionistic set theory in which there exists an infinite set A, comprising the naturals, that is, in one-one correspondence with the set of all functions from A to A, thereby furnishing a set-theoretic model of the l-calculus, that is, one in which the l-terms are interpreted as set-theoretic functions. We think that these realisability universes lend themselves to further interesting applications in the metamathematics of intuitionistic set theories.