Extensional realizability and choice for dependent types in intuitionistic set theory

Abstract

In "Extensional realizability for intuitionistic set theory", we introduced an extensional variant of generic realizability, where realizers act extensionally on realizers, and showed that this form of realizability provides "inner" models of $\mathsf{CZF}$ (constructive Zermelo-Fraenkel set theory) and $\mathsf{IZF}$ (intuitionistic Zermelo-Fraenkel set theory), that further validate ${\mathsf{AC}}_{\mathsf{FT}}$ (the axiom of choice in all finite types). In this paper, we show that extensional generic realizability validates several choice principles for dependent types, all exceeding ${\mathsf{AC}}_{\mathsf{FT}}$. We then show that adding such choice principles does not change the arithmetic part of either $\mathsf{CZF}$ or $\mathsf{IZF}$.