In Chapter 14 of Jech's Set Theory, he defines $P$-names in terms of of a certain Boolean-valued model and the forcing relation in terms of these names, as shown below:
Let $M$ be a transitive model of ZFC (the ground model), and let $(P,<)\in M$ be a notion of forcing. By Corollary $14.12$ there exists a complete Boolean algebra $B=B(P)$ such that $P$ embeds in $B$ by a mapping $e:P\to B$ that satisfies (14.6). We use $M^B$ to denote the $B$-valued model defined in (14.15) (inside $M$).
Definition 14.26. $M^P=M^{B(P)}$. The elements of $M^P$ are called $P$–names, (or just names). $P$-names are usually denoted by dotted letters. The forcing relation $\Vdash_P$ (or just $\Vdash$) is defined by
$$p\Vdash \varphi(\dot a_1,\dots,\dot a_n) \text{ if and only if } e(p)\leq ||\varphi(\dot a_1,\dots,\dot a_n)||,$$
where $\varphi$ is a formula of set theory and $\dot a_1,\dots,\dot a_n$ are names.
He then goes on to define a "name for $M$" and the "canonical name for a generic filter on $P$" as follows:
Among $P$-names there are the canonical names $\hat x$ (it's actually an inverted hat but I don't know how to type that in LaTeX so bear with me; if you happen to know I would greatly appreciate some help).
We can also introduce a "name for $M$;" since $a\in M\leftrightarrow(\exists x\in M)a=x$, we define
$$(14.28)\qquad p\Vdash \dot a\in \hat M\text{ if and only if } \forall q\leq p \exists r\leq q \exists x(r\Vdash \dot a =\hat x).$$
Finally, we consider the canonical name for a generic filter on $P$. Using Definition 14.25 for $B(P)$ and the relation between generic filters on $P$ and generic ultrafilters on $B(P)$ spelled out in Lemma 14.13, we arrive at the following definition:
$$(14.29) \qquad p\Vdash q\in \dot G \text{ if and only if } \forall r\leq p\exists s\leq r \, s\leq q.$$
I have a one hopefully quick clarification question, and then a more substantial question.
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When Jech writes "We use $M^B$ to denote the $B$-valued model defined in (14.15) (inside $M$)," does he just mean $V^B\cap M$? Or are we doing this process "inside $M$" in some different way? You can see (14.15) in my other question here.
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I think I understand (14.28) is saying in terms of the forcing relation, I don't really understand how this provides a "name for $M$." Could someone possibly elaborate on what Jech means by that? The same question applies to (14.29), but I feel like answering one will answer the other.
Best Answer
Question 1: $M^B$ is obtained by taking the whole definition of $V^B$, for complete Boolean algebras $B$ in $V$, and interpreting everything in $M$. It is what $M$ thinks is the $B$-valued universe.
I'd rather not think in terms of $V^B$, as in your suggestion of $V^B\cap M$, because it's not clear what $V^B$ would mean here. The problem is that $B$ is a complete Boolean algebra in the sense of $M$, i.e., its subsets in $M$ have suprema and infima, but it is usually not complete in $V$.
Question 2: The idea that $\check M$ serves as a name for the ground model is justified after the fact, when one proves that, for any name $\dot a$, the interpretation of $\dot a$ with respect to a generic $G$ is in the ground model $M$ iff $\Vert\dot a\in \check M\Vert\in G$.
Similarly, one checks later that the interpretation of $\dot G$ with respect to any generic $G$ is $G$ itself.
You might find it useful to check that the truth value of "$\dot G$ is an $\check M$-generic filter in $\check P$" is $1$.