Worked solution: Gödel and Cohen: the continuum hypothesis is independent of ZFC (1963)

By transfinite recursion on the ordinals: $L_0 = \emptyset$, $L_{\alpha+1} = \mathrm{Def}(L_\alpha)$ (the collection of all subsets of $L_\alpha$ that are first-order definable, with parameters, over the structure $\langle L_\alpha, \in \rangle$), $L_\lambda = \bigcup_{\alpha<\lambda} L_\alpha$ at limit ordinals $\lambda$, and $L = \bigcup_{\alpha \in \mathrm{Ord}} L_\alpha$.

Gödel proved in 1938–40 that $L$ satisfies every $\mathrm{ZFC}$ axiom, and in fact satisfies the stronger axiom of constructibility $V = L$ ("every set is constructible"). Because $L$ is built purely from definable steps, it is far more rigid and controllable than the full universe $V$ of all sets could ever be assumed to be.

Having a rigid, well-understood universe like $L$ is only useful if one can actually compute inside it what cardinalities look like; the next step shows how a structural fact about $L$ (the condensation lemma) pins down the size of $\mathcal{P}(\omega) \cap L$ exactly, which is what will finally deliver $\mathrm{CH}$.