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  "body": "\n## Definition\n\nA *forcing notion* is a [[partial order|0QV2JS9RJ3]] $(P, \\le)$ with a greatest element $1$.\nIts elements are called *conditions*, and $q \\le p$ is read \"$q$ is stronger than $p$\".\n\nLet $M$ be a countable transitive model of [[ZFC|0HE5WDQ9FJ]] with $P \\in M$.\nA *$P$-name* is a set $\\tau$ of pairs $(\\sigma, p)$, where $\\sigma$ is a $P$-name and $p \\in P$; this is a definition by recursion on rank.\nFor a filter $G \\subseteq P$, the *value* of a name is defined by recursion:\n$$\\tau^G = \\set{\\sigma^G : (\\sigma, p) \\in \\tau \\text{ for some } p \\in G}.$$\nFor a [[generic filter|WEDP73Y7GK]] $G$ over $M$, the *forcing extension* is $M[G] = \\set{\\tau^G : \\tau \\in M \\text{ is a } P\\text{-name}}$.\nLet $p \\in P$, let $\\varphi(x_1, \\ldots, x_n)$ be a formula, and let $\\tau_1, \\ldots, \\tau_n \\in M$ be names.\nThe condition $p$ *forces* $\\varphi(\\tau_1, \\ldots, \\tau_n)$, written $p \\Vdash \\varphi(\\tau_1, \\ldots, \\tau_n)$, if $M[G] \\models \\varphi(\\tau_1^G, \\ldots, \\tau_n^G)$ for every generic $G$ over $M$ with $p \\in G$.\n\n## Remarks\n\nThe basic theorem of forcing says that $M[G]$ is a countable transitive model of ZFC, that $M \\subseteq M[G]$ and $G \\in M[G]$, and that $M$ and $M[G]$ have the same ordinals.\nThe forcing theorem says that a statement holds in $M[G]$ if and only if some $p \\in G$ forces it.\nIt also says that the relation $\\Vdash$ restricted to each formula is definable in $M$.\nExample: *Cohen forcing* is the set of finite partial functions from $\\omega$ to $\\set{0, 1}$, ordered by reverse inclusion, with $1 = \\emptyset$.\nFor a generic $G$, the union $\\bigcup G$ is a function from $\\omega$ to $\\set{0, 1}$ that is not in $M$.\nA non-example of a forcing notion is a partial order without a greatest element; this is a minor point, since one can add a new top element.\nSome authors use preorders instead of partial orders, or write $p \\le q$ for \"$p$ is weaker\", so the notation must be checked in each source.\n",
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