I have seen the proof of transitioning from a model in which CH holds to a model in which CH fails. However, how do we force the other direction? More explicitly, suppose that $M$ is a countable, transative model of ZFC and $M \models \neg$ CH. What poset $\mathbb{P} \in M$ do we use to find a generic filter $G$ over $\mathbb{P}$ such that $M[G] \models$ CH?
Thanks.
We can always collapse the continuum to be $\aleph_1$. Namely, $\Bbb P$ is the partial order whose conditions are countable partial functions from $\omega_1$ to $\Bbb R$, ordered by reverse inclusion (or inclusion, if you work by the Jerusalem convention).
Genericity arguments show that the generic defines a surjection from $\omega_1$ onto the real numbers, but this forcing is also countably closed so it adds no new real numbers. Therefore $\sf CH$ holds in the extension.