A formalization of forcing and the unprovability of the continuum hypothesis

TL;DR

This paper formalizes the method of forcing within the Lean 3 theorem prover, demonstrating its application by verifying the unprovability of the continuum hypothesis in a specific Boolean-valued model.

Contribution

It introduces a formalization of forcing in Lean 3, including core theorems and a deep embedding of logic, enabling rigorous verification of set-theoretic independence results.

Findings

Formalization of forcing in Lean 3

Verification of the failure of the continuum hypothesis

Deep embedding of first-order logic with Boolean-valued soundness

Abstract

We describe a formalization of forcing using Boolean-valued models in the Lean 3 theorem prover, including the fundamental theorem of forcing and a deep embedding of first-order logic with a Boolean-valued soundness theorem. As an application of our framework, we specialize our construction to the Boolean algebra of regular opens of the Cantor space $2^{\omega_2 \times \omega}$ and formally verify the failure of the continuum hypothesis in the resulting model.