Heyting-valued interpretations for Constructive Set Theory

The theory of locales [23] has a twofold interplay with intuitionistic mathematics: first of all, the internal logic of toposes and intuitionistic set theories provide suitable settings for the development of the theory of locales [24], and secondly, the notion of a locale determines two important forms of toposes and of interpretations for intuitionistic set theories, namely localic toposes [26, Chapter IX] and Heyting-valued interpretations [10]. The combination of these two aspects has led to many proof-theoretic applications [16, 17] and important results in the theory of elementary toposes [25]. The internal logic of toposes with a natural number object [9] and intuitionistic set theories [34] are examples of formal systems that are fully impredicative, in the sense that they have proof-theoretic strength above the one of second-order arithmetic [5]. Formal topology originated by considering whether it was possible to develop pointfree topology in a generalised predicative context [30]. Generalised predicative mathematics is understood here as something more general than the Weyl-Feferman-Schütte notion of predicative mathematics, so as to allow generalised inductive definitions and generalised reflection [15, 29]. For instance, Martin-Löf type theories with well-ordering types and Mahlo universe types are generalised predicative systems, and so is every formal system that is proof-theoretically reducible to them. By virtue of the type-theoretic interpretation [2, 3, 4], the constructive set theories that we consider here are generalised predicative systems. The development of formal topology shows that it is possible to reconstruct considerable parts of pointfree topology within Martin-Löf type theories [31]. Yet, the second aspect of relationship between locale theory and