# De Jongh's Theorem for Intuitionistic Zermelo-Fraenkel Set Theory

## TL;DR

This paper proves that the propositional logic of intuitionistic set theory IZF aligns with IPC and establishes the de Jongh property for IZF and CZF relative to certain intermediate logics, advancing understanding of their logical foundations.

## Contribution

It demonstrates that IZF and CZF have the de Jongh property with respect to a broad class of intermediate logics, extending previous results in intuitionistic set theory.

## Key findings

- IZF's propositional logic is IPC
- IZF has the de Jongh property for certain intermediate logics
- Results apply similarly to CZF

## Abstract

We prove that the propositional logic of intuitionistic set theory IZF is intuitionistic propositional logic IPC. More generally, we show that IZF has the de Jongh property with respect to every intermediate logic that is complete with respect to a class of finite trees. The same results follow for CZF.