Models of Intuitionistic Set Theory in Subtoposes of Nested Realizability Toposes

TL;DR

This paper constructs models of intuitionistic set theory within nested realizability toposes and their subtoposes, extending the framework to various realizability toposes including relative and Herbrand toposes.

Contribution

It introduces a method to associate classes of small maps in nested realizability toposes and their subtoposes, enabling models of intuitionistic set theory in these structures.

Findings

Models of intuitionistic set theory are constructed in nested realizability toposes.

The approach applies to various realizability toposes, including relative and Herbrand toposes.

A systematic method for inducing small maps in subtoposes is developed.

Abstract

With every pca $\mathcal{A}$ and subpca $\mathcal{A}_\#$ we associate the nested realizability topos $\mathsf{RT}(\mathcal{A},\mathcal{A}_\#)$ within which we identify a class of small maps $\mathcal{S}$ giving rise to a model of intuitionistic set theory within $\mathsf{RT}(\mathcal{A},\mathcal{A}_\#)$. For every subtopos $\mathcal{E}$ of such a nested realizability topos we construct an induced class $\mathcal{S_E}$ of small maps in $\mathcal{E}$ giving rise to a model of intuitionistic set theory within $\mathcal{E}$. This covers relative realizability toposes, modified relative realizability toposes, the modified realizability topos and van den Berg's recent Herbrand topos.