In their paper "Pointwise Definable Models of Set Theory" Hamkins, Linetsky, and Reitz prove the following theorem:
"Every countable model of ZFC has a pointwise definable class forcing extension."
Let's consider, as our ground model, a c.t.m.--call it M-- that satisfies CH. By the theorem stated above, M has a pointwise definable class forcing extension M[G]. Can one provide a relatively simple example of such an M[G] where CH is false? Are there any limitations (apart from Koenig's Theorem) as to how such an M[G] can violate CH? Also, as regards set theorists living (so to speak) in M[G], do they believe that the language of set theory that describes their set-theoretic universe M[G] is a countable language?
The argument in the paper proceeds in two steps: adding a special predicate $U$ which makes the structure $(M,\in,U)$ pointwise definable and then forcing over this to code the whole structure into the GCH pattern. The first step doesn't depend at all on the GCH pattern of $M$ and the second step can be taken to have arbitrarily high closure and so it can be made to preserve arbitrarily long initial segments of the GCH pattern of $M$. So to get a pointwise definable $M[G]$ with a specified violation of CH we just start with $M$ satisfying CH, force over it to get the desired value of $2^{\aleph_0}$ and then perform the argument from the paper above this value. In particular, any value for the continuum which is consistent with König's theorem is also consistent with the universe being pointwise definable.