A Normalizing Intuitionistic Set Theory with Inaccessible Sets

TL;DR

This paper introduces extit{IZFio}, a robust set theory with inaccessible sets that supports interpreting advanced type theories, enabling proof assistants and program extraction from constructive proofs.

Contribution

It axiomatizes an impredicative constructive set theory with inaccessible sets, bridging set theory and type theory for proof assistants like LEGO and Coq.

Findings

Defines a new set theory extit{IZFio} with inaccessible sets

Establishes normalization via realizability for the associated lambda calculus

Enables program extraction from constructive proofs in the set theory

Abstract

We propose a set theory strong enough to interpret powerful type theories underlying proof assistants such as LEGO and also possibly Coq, which at the same time enables program extraction from its constructive proofs. For this purpose, we axiomatize an impredicative constructive version of Zermelo-Fraenkel set theory IZF with Replacement and $\omega$-many inaccessibles, which we call \izfio. Our axiomatization utilizes set terms, an inductive definition of inaccessible sets and the mutually recursive nature of equality and membership relations. It allows us to define a weakly-normalizing typed lambda calculus corresponding to proofs in \izfio according to the Curry-Howard isomorphism principle. We use realizability to prove the normalization theorem, which provides a basis for program extraction capability.