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?