When do Generic sets exist?

Question

In the proof of the independence of the continuum hypothesis in Jech's 'Set Theory', the author goes through the standard proof procedure

dom$(p)$ is a finite subset of $\omega_2\times \omega$

ran$(p)\subset(0,1)$

and let $p$ be stronger than $q$ if $q\subset p$. This gives us a poset $(P,<)$.

Then Jech says

'Let $G$ be a generic set of conditions.'

Now, can we guarantee that such a $G$ actually exists? Earlier in the chapter Jech shows that if $D$ is a countable collection of subsets of a poset $P$, then a $D$-generic filter exists. However in this case even if we take our model to be countable, the number of dense subsets of $P$ could surely be uncountable? I'm not sure what I'm missing here.

Answer 1

The set $D$ is taken to consist of the dense subsets of $P$ that are elements of the countable ground model, so $D$ is countable.