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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partial order|0QV2JS9RJ3]], and let $M$ be a set or class.\nA subset $D \\subseteq P$ is *dense* if for every $p \\in P$ there is $q \\in D$ with $q \\le p$.\nA subset $G \\subseteq P$ is a *filter* if:\n\n1. $G$ is nonempty.\n2. If $p \\in G$ and $p \\le q$, then $q \\in G$.\n3. If $p, q \\in G$, then there is $r \\in G$ with $r \\le p$ and $r \\le q$.\n\nA filter $G$ is *generic over $M$*, or *$M$-generic*, if $G \\cap D \\neq \\emptyset$ for every dense set $D \\subseteq P$ with $D \\in M$.\n\n## Remarks\n\nIf $P$ has a least element $0$, then $G = P$ is generic over every $M$: each dense set $D$ contains some $q \\le 0$, so $0 \\in D$.\nNon-example: let $P$ be the set of finite partial functions from $\\omega$ to $\\set{0, 1}$, ordered by reverse inclusion.\nLet $M$ be a transitive model of ZFC with $P \\in M$, and let $G \\in M$ be a filter.\nThe set $D = P \\setminus G$ is in $M$, and it is dense.\nGiven $p$, 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 $n$.\nSo $G \\cap D = \\emptyset$, and $G$ is not generic over $M$.\nIf $M$ is countable, the Rasiowa–Sikorski lemma gives, for each $p \\in P$, a generic filter over $M$ that contains $p$.\nSome authors also allow open dense sets or maximal antichains in the definition; these give the same notion of genericity.\n",
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