Given a model $M$ of ZFC, we can define $\mathbb{P}$-names and generic extensions $M[G]$ in terms of $M$.
But the usual framework for forcing involves an outer model $M\subseteq V$ of ZFC and $M$ is required to be transitive and countable with respect to $V$. The forcing extensions then become $M\subseteq M[G] \subseteq V$.
Does anyone know why this outer model $V$ is important for forcing? Extending $M$ looks enough to prove independances of axioms such as Choice or the Continuum Hypothesis.
EDIT
Following the answer below, here is a tentative to construct a generic filter on $M$ without an outer model $V$.
We want to prove the relative consistency of a new axiom of set theory. So we start by assuming that ZFC is consistent. By the Lowenheim-Skolem theorem, there is a countable model $M$ of ZFC. Let $(\mathbb{P}, \leq)$ a partial order inside $M$ of conditions that represent all the possibilities of the forcing extension.
Because $M$ is countable, so are the $\mathbb{P}$-dense parts, that we enumerate $D_0,D_1,\dots$ Let $p_0\in D_0$. By density there is $p_1\in D_1$ such as $p_1\leq p_0$. We continue this to define a decreasing sequence $p_n\in D_n$. Then we call $G=\{p\in\mathbb{P} | \exists i\in\mathbb{N}, p_i\leq p\}$. $G$ is a filter for $\leq$ and it meets every dense part, so it is $M$-generic on $\mathbb{P}$.
Is there a problem with this construction, that does not even need that $M$ is transitive?
Best Answer
The problem comes with this step:
"Letting $G$ be a $\mathbb{P}$-generic filter over $M$, we define $M[G]$ as [stuff]."
How do we know that such a $G$ exists in the first place? Certainly it will not in general from "within $M$:" as long as $\mathbb{P}$ is nontrivial, no $\mathbb{P}$-generic filter over $M$ will exist in $M$. Generic filters over $M$ have to come from some larger model.
This is one of the reasons that the Boolean-valued models approach to forcing is enticing: everything takes place within the original model, and we can talk about forcing over the "real" universe without saying horrible nonsense.