Choice and independence of premise rules in intuitionistic set theory

Abstract

Choice and independence of premise principles play an important role in characterizing Kreisel's modified realizability and Gödel's Dialectica interpretation. In this paper we show that a great many intuitionistic set theories are closed under the corresponding rules for finite types over $\mathbb{N}$. It is also shown that the existence property (or existential definability property) holds for statements of the form $\exists y^{\sigma }\phi (y)$, where the variable $y$ ranges over objects of finite type $\sigma$. This applies in particular to $\mathsf{CZF}$ (Constructive Zermelo-Fraenkel set theory) and $\mathsf{IZF}$ (Intuitionistic Zermelo-Fraenkel set theory), two systems known not to have the general existence property. On the technical side, the paper uses a method that amalgamates generic realizability for set theory with truth, whereby the underlying partial combinatory algebra is required to contain all objects of finite type.