Abstract
In [8] 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 IZF. Here we show that this can be extended to a much wider class of realizability toposes as considered in [4] and [3].For this purpose we first show this result for nested realizability toposesRT(A,A#) induced by a pca A together with a sub-pca A# as considered implicitly in [4] and then show that it is preserved by restriction to subtoposes. This suffices since all toposes considered in [4] and [3] arise as subtoposes of some nested realizability toposes.
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