# forcing relation

If $\mathfrak{M}$ is a transitive model of set theory and $P$ is a partial order then we can define a forcing relation:

 $p\Vdash_{P}\phi(\tau_{1},\ldots,\tau_{n})$

($p$ 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 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 $G\subseteq P$, $\mathfrak{M}[G]\vDash\phi$.

Title forcing relation ForcingRelation 2013-03-22 12:53:28 2013-03-22 12:53:28 Henry (455) Henry (455) 5 Henry (455) Definition msc 03E35 msc 03E40 Forcing forcing relation forces