TITLE: Forcing over an arbitrary model of ZFC
QUESTION [14 upvotes]: I’m learning set theory and forcing in the (french) book from Jean-Louis Krivine “Théorie des ensembles”.
Given a countable transitive model $\mathscr{M}$ of ZFC together with a poset $P$, he constructs the model $\mathscr{M}[G]$ where $G$ is a $P$-generic in the ambient universe $\mathscr{U}$. The countability of $\mathscr{M}$ is essential in order for such a generic $G$ to exist.
But I saw several times in answers on MO that forcing could be defined over any model of ZFC and that, for example, CH and ~CH can both be forced over any model of ZFC.
My questions are:

How does this kind of forcing work? I guess that we cannot anymore suppose that there exists a generic $G$, so how is the new universe constructed? And what does the truth lemma becomes?
When do we need to use this kind of forcing? For example if I want to prove that some proposition $P$ is independent of ZFC, I can always assume that my initial model of ZFC is countable and “usual” forcing will probably be sufficient.

Any reference about this will be more than welcome.

REPLY [6 votes]: $\def\zfc{\mathit{ZFC}}$No one seems to have mentioned another way to think about forcing, that can be sometimes helpful.
Let $\zfc_B$ denote $\zfc$ extended with an additional constant $B$ and an axiom that $B$ is a complete Boolean algebra. Let $\zfc_{B,G}$ be a theory in the language $\langle\in,B,G,U\rangle$ where $G$ is a constant, $U$ a unary predicate, with axioms stating that $U$ defines a transitive class, $\zfc_B$ holds in $U$, $\zfc$ (including replacement for all the new stuff) holds in the whole universe $V$, $G$ is a $U$-generic filter on $B$, and $V=U[G]$.
Any method of making sense of forcing (such as in the answers above) shows that $\zfc_{B,G}$ is a conservative extension of $\zfc_B$, in the sense that if $\phi$ is a $\langle\in,B\rangle$-sentence such that $\zfc_{B,G}\vdash\phi^U$, then $\zfc_B\vdash\phi$. Note that this can even be shown purely syntactically in a very weak metatheory.
In model-theoretic terms, this means that if $M\models\zfc$ and $M$ thinks that $B\in M$ is a complete Boolean algebra, then $M$ has an elementary extension $M'$ which has a generic extension $M'[G]$ with a generic for $B$.
However, the main point is that if you are now working inside $\zfc$ and you come upon a complete Boolean algebra $B$, you can safely pretend that your universe $V$ is included in a generic extension $V[G]$, even though you cannot actually define it from inside.