A Formal Proof of the Independence of the Continuum Hypothesis

We describe a formal proof of the independence of the continuum hypothesis ($\mathsf{CH}$) in the Lean theorem prover. We use Boolean-valued models to give forcing arguments for both directions, using Cohen forcing for the consistency of $\neg \mathsf{CH}$ and a $\sigma$-closed forcing for the consistency of $\mathsf{CH}$.