Extensional realizability for intuitionistic set theory

Abstract

Abstract In generic realizability for set theories, realizers treat unbounded quantifiers generically. To this form of realizability, we add another layer of extensionality by requiring that realizers ought to act extensionally on realizers, giving rise to a realizability universe $\mathrm{V_{ex}}(A)$ in which the axiom of choice in all finite types, ${\textsf{AC}}_{{\textsf{FT}}}$, is realized, where $A$ stands for an arbitrary partial combinatory algebra. This construction furnishes ‘inner models’ of many set theories that additionally validate ${\textsf{AC}}_{{\textsf{FT}}}$, in particular it provides a self-validating semantics for ${\textsf{CZF}}$ (constructive Zermelo–Fraenkel set theory) and ${\textsf{IZF}}$ (intuitionistic Zermelo–Fraenkel set theory). One can also add large set axioms and many other principles.