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 independant 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.

Thanks.

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.