Solving Recursive Domain Equations in Models of Intuitionistic Set Theory

Synopsis We give a general axiomatic construction of solutions to recursive domain equations, applicable both to classical models of domain theory and to realizability models. The approach is based on embedding categories of predomains in models of intuitionistic set theory. We show that the existence of solutions to recursive domain equations depends on the strength of the set theory. Such solutions do not exist in general when predomains are embedded in an elementary topos. They do exist when predomains are embedded in a model of Intuitionistic Zermelo-Fraenkel set theory, in which case we give a fibrational account of algebraic compactness.