Generalizing realizability and Heyting models for constructive set theory

This article presents a common generalization of the two main methods for obtaining class models of constructive set theory. Heyting models are a generalization of the Boolean models for classical set theory which are a variant of forcing, while realizability is a decidedly constructive method that has first been developed for number theory by Kleene and was later very fruitfully adapted to constructive set theory. In order to achieve the generalization, a new kind of structure (applicative topologies) is introduced, which contains both elements of formal topology and applicative structures. This approach not only deepens the understanding of class models and leads to more efficiency in proofs about these kinds of models, but also makes it possible to prove new results about the two special cases that were not known before and to construct new models.

► Realizability and Heyting constructions provide models for constructive set theory.
► We work in the predicative framework of CZF.
► We propose a common generalization of the two constructions.
► This can be used to construct new models and generalize results.