The formal verification of the ctm approach to forcing

We discuss some highlights of our computer-verified proof of the construction, given a countable transitive set-model $M$ of $ZFC$, of generic extensions satisfying $ZFC+\mathrm{\neg }CH$ and $ZFC+CH$. Moreover, let $\mathcal{R}$ be the set of instances of the Axiom of Replacement. We isolated a 21-element subset $\mathrm{\Omega }\subseteq \mathcal{R}$ and defined $\mathcal{F}:\mathcal{R}\to \mathcal{R}$ such that for every $\mathrm{\Phi }\subseteq \mathcal{R}$ and $M$-generic $G$, $M\models ZC\cup \mathcal{F}\text{“}\mathrm{\Phi }\cup \mathrm{\Omega }$ implies $M[G]\models ZC\cup \mathrm{\Phi }\cup \{\mathrm{\neg }CH\}$, where $ZC$ is Zermelo set theory with Choice. To achieve this, we worked in the proof assistant Isabelle, basing our development on the Isabelle/ZF library by L. Paulson and others.