Models of Intuitionistic Set Theory in Subtoposes of Nested Realizability Toposes

Joyal and Moerdijk have shown that realizability toposes over partial combinatory algebras (pca) host classes of small maps giving rise to initial ZF-algebras providing models of intuitionistic Zermelo–Fraenkel set theory. Here we show that this can be extended to a much wider class of realizability toposes. For this purpose we first show this result for nested realizability toposes induced by a pca together with a sub-pca and then show that it is preserved by restriction to subtoposes.