Range of a P-name

Question

I am working on a set theory problem from Kunen's Set Theory book, and it involves knowing $\text{ran}(\tau)$ where $\tau$ is a $\mathbb{P}$-name. The entire section loves to talk about the domain of things like $\tau$, but not the range. Is $$\text{ran}(\tau) = \{p \in \mathbb{P} : \exists \sigma \in M^{\mathbb{P}}(\langle \sigma, p \rangle \in \tau)\}$$ correct? I don't think I have the right thing since this doesn't seem to help me solve my problem.

Answer 1

The name $\tau\in M^\mathbb{P}$ is an element of the set $M$, and in particular, it is a relation. So you can compute its range $$ \mathrm{ran}(\tau) = \{p\in \mathbb{P} : \exists \sigma\,(\langle\sigma,p\rangle\in\tau)\}. $$

But in this particular problem, $\tau$ is the name of a function and you have the expression $$ p\Vdash \check b\in\mathrm{ran}(\tau). $$

So you have to read the expression to the right of $\Vdash$ as a formula of the forcing language. Therefore, $p$ forces that $b$ belong to the range of the function denoted by $\tau$. In other words, using the Fundamental Theorem, $$ M[G]\models b\in \mathrm{ran}(\tau_G) $$ for every generic such that $p\in G$ and $b\in M$.