The Relative Consistency of the Axiom of Choice Mechanized Using Isabelle/ZF

TL;DR

This paper presents a formal mechanization of the proof of the relative consistency of the axiom of choice using Isabelle/ZF, highlighting the challenges and importance of formalizing metatheory in deep mathematical proofs.

Contribution

It provides a comprehensive mechanization of the consistency proof, addressing metatheoretical complexities and demonstrating the feasibility of formalizing advanced set theory.

Findings

Mechanization of the consistency proof is possible with Isabelle/ZF.

Formalizing metatheory is essential for deep mathematical proofs.

The development supports further formalization of set theory from Kunen's textbook.

Abstract

The proof of the relative consistency of the axiom of choice has been mechanized using Isabelle/ZF. The proof builds upon a previous mechanization of the reflection theorem. The heavy reliance on metatheory in the original proof makes the formalization unusually long, and not entirely satisfactory: two parts of the proof do not fit together. It seems impossible to solve these problems without formalizing the metatheory. However, the present development follows a standard textbook, Kunen's Set Theory, and could support the formalization of further material from that book. It also serves as an example of what to expect when deep mathematics is formalized.