A formalization of forcing and the unprovability of the continuum hypothesis

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.