Dense open sets and generic filters over a model

Use Forcing preorders, compatibility and filters: $q\le p$ means stronger, and a filter is nonempty, upward closed and internally downward directed. A subset $D\subseteq P$ is dense if $\mathrm{\forall }p\in P\mathrm{\exists }q\in D(q\le p)$, and open if $p\in D$ and $q\le p$ imply $q\in D$.

For a transitive ZF model M containing P and its order, a filter $G\subseteq P$ is M-generic if $G\cap D\mathrm{\ne }\mathrm{\varnothing }$ for every dense $D\subseteq P$ with $D\in M$. The displayed density condition is the same inside and outside M: all its quantifiers range over the identical sets P and D, with identical order relation. No largest condition, countability or existence of a generic filter is assumed by the definition. When M is called countable, this means countable externally; its natural numbers agree with actual omega by Ordinals and omega in transitive models.