Forcing name ($\tau \in M^{\mathbb{P}}$)

A forcing name is a recursively built set of pairs $(\sigma ,p)$, where $\sigma$ is a lower-rank forcing name and $p$ is a forcing condition. Ground-model names are interpreted using a generic filter to form the generic extension.

The forcing name $\{(\check{p},q):p,q\in \mathbb{P},q\perp p\}$ evaluates to $\mathbb{P}\setminus G$. If $p\notin G$, genericity applied to $\{q:q\le p\text{ or }q\perp p\}$ supplies an incompatible member of $G$. If $p\in G$, directedness prevents such a member.

Interpret a forcing name recursively by $\mathrm{val}(\tau ,G)=\{\mathrm{val}(\sigma ,G):\exists p\in G((\sigma ,p)\in \tau )\}$. The collection of interpreted ground-model names is the generic extension.

The canonical forcing name for a ground-model set $x$ is $\check{x}=\{(\check{y},1):y\in x\}$. Its evaluation of a forcing name under every generic filter is $x$.