Extensional realizability and choice for dependent types in intuitionistic set theory

TL;DR

This paper extends extensional realizability to validate stronger choice principles for dependent types in intuitionistic set theory, without affecting the core arithmetic of the theories.

Contribution

It demonstrates that extensional generic realizability can validate stronger choice principles for dependent types, surpassing ${ m AC}_{ m FT}$, while preserving the arithmetic core of $ extsf{CZF}$ and $ extsf{IZF}$.

Findings

Validated several choice principles for dependent types exceeding ${ m AC}_{ m FT}$.

Adding these principles does not alter the arithmetic part of $ extsf{CZF}$ or $ extsf{IZF}$.

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 $\sf CZF$ (constructive Zermelo-Fraenkel set theory) and $\sf IZF$ (intuitionistic Zermelo-Fraenkel set theory), that further validate ${\sf AC}_{\sf FT}$ (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 ${\sf AC}_{\sf FT}$. We then show that adding such choice principles does not change the arithmetic part of either $\sf CZF$ or $\sf IZF$.