Generic filter

Remarks

If $P$ has a least element $0$, then $G = P$ is generic over every $M$: each dense set $D$ contains some $q \le 0$, so $0 \in D$. Non-example: let $P$ be the set of finite partial functions from $\omega$ to $\set{0, 1}$, ordered by reverse inclusion. Let $M$ be a transitive model of ZFC with $P \in M$, and let $G \in M$ be a filter. The set $D = P \setminus G$ is in $M$, and it is dense. Given $p$, pick $n \notin \operatorname{dom} p$, and extend $p$ at $n$ by a value that disagrees with $\bigcup G$ at $n$, or by any value if $\bigcup G$ is undefined at $n$. So $G \cap D = \emptyset$, and $G$ is not generic over $M$. If $M$ is countable, the Rasiowa–Sikorski lemma gives, for each $p \in P$, a generic filter over $M$ that contains $p$. Some authors also allow open dense sets or maximal antichains in the definition; these give the same notion of genericity.