forcing relationForcingRelation2002-07-30 20:21:082002-07-31 20:39:16Definitionforcing relationforces% this is the default PlanetMath preamble. as your knowledge
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%\PMlinkescapeword{theory}If $\mathfrak{M}$ is a transitive model of set theory and $P$ is a partial order then we can define a \emph{forcing relation}:
$$p\Vdash_P \phi(\tau_1,\ldots,\tau_n)$$
($p$ \emph{forces} $\phi(\tau_1,\ldots,\tau_n)$)
for any $p\in P$, where $\tau_1,\ldots,\tau_n$ are $P$- names.
Specifically, the relation holds if for every generic filter $G$ over $P$ which contains $p$, $$\mathfrak{M}[G]\vDash \phi(\tau_1[G],\ldots,\tau_n[G])$$
That is, $p$ forces $\phi$ if every \PMlinkescapetext{extension} of $\mathfrak{M}$ by a generic filter over $P$ containing $p$ makes $\phi$ true.
If $p\Vdash_P \phi$ holds for every $p\in P$ then we can write $\Vdash_P\phi$ to mean that for any generic $G\subseteq P$, $\mathfrak{M}[G]\vDash\phi$.