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