The first-order logic of CZF is intuitionistic first-order logic

TL;DR

This paper demonstrates that the first-order logic of Constructive Zermelo-Fraenkel set theory (CZF) aligns with intuitionistic first-order logic by introducing a novel model of transfinite computation and analyzing admissible rules.

Contribution

It introduces Set Register Machines as a new model of transfinite computation and establishes the equivalence of CZF's first-order logic with intuitionistic logic.

Findings

The first-order logic of CZF is intuitionistic.

Propositional admissible rules of CZF match those of intuitionistic propositional logic.

Set Register Machines provide a new framework for realisability in set theory.

Abstract

We prove that the first-order logic of CZF is intuitionistic first-order logic. To do so, we introduce a new model of transfinite computation (Set Register Machines) and combine the resulting notion of realisability with Beth semantics. On the way, we also show that the propositional admissible rules of CZF are exactly those of intuitionistic propositional logic.