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

Defines:

forcing relation, forces

Related:

Forcing

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

03E35*no label found*03E40

*no label found*

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