Apologies if this is a dumb question - I’m trying to dig into the intricacies of set-theoretic forcing at the moment and am struggling with a step.
In particular, given our poset P we need to show that our generic ultra-filter G does not already exist in our transitive model M. I think I understand the argument around this - we can show the subset P\G is also dense and clearly G cannot intersect this, so no such G exists in our model on pain of inconsistency
My (probably naive) confusion is about why, when we build our extended model M[G], can we not run a similar argument to show there is a dense subset of P that G does not intersect? My intuition is that the “generic-ness” of G is not a property we require G to have in M[G], but rather a property we require G to have to guarantee it does not exist in M (and that it has certain filter properties etc), and once we extend M to include G we no longer care whether it intersects every dense subset of P in M[G]
Is this intuition correct or am I missing something fundamental in the argument?
Yes, $G$ meets every dense set in $M$ but not in $M[G]$. So $G$ is $\mathbb{P}$-generic over $M$, but not $\mathbb{P}$-generic over $M[G]$, and that's fine.
(Incidentaly, it's "generic filter," not "generic ultrafilter," in the context of forcing-via-posets; ultrafilters emerge when we use the Boolean algebras approach to forcing.)