TITLE: When does the choice of the generic matter?
QUESTION [26 upvotes]: It is a somewhat curious phenomenon that, in forcing arguments, one usually doesn't care about any particular properties of the generic filter being used (this isn't strictly true; there are cases where we force below some sort of master condition, for example, but this basically amounts to asking my question for the cone below the condition). This position might possibly be preferred by mathematicians who interpret talking about generic filters as "semantic sugar" for purely syntactic arguments about Boolean truth values, but if we are prepared to talk about generics as real objects, it seems weird that we don't pay very much attention to the structure of the objects which generate our extensions.
But perhaps we needn't always care about the particular generic. Let $M$ be a transitive model of set theory. Call a notion of forcing $\mathbb{P}\in M$ forcing agnostic (over $M$) if for any two $M$-generic filters $G,H\subseteq \mathbb{P}$ the two extensions $M[G]$ and $M[H]$ are elementarily equivalent.
There is an obvious reformulation of forcing agnosticism: $\mathbb{P}$ is forcing agnostic iff the Boolean value, with respect to the Boolean algebra associated to $\mathbb{P}$, of any sentence (without parameters in the forcing language) is either 0 or 1. This immediately implies that any almost homogeneous forcing is forcing agnostic; in fact, any two extensions by almost homogeneous forcing are elementarily equivalent in the language augmented with constants for elements of the ground model.
Is there a characterization of forcing agnostic posets? Is this a purely structural property of the poset or does the ambient model matter, i.e. can  a poset be forcing agnostic over some models but not over others?
I would also welcome any examples of forcing arguments where some care is needed in choosing the generic (in addition to ensuring a particular condition gets in).

REPLY [7 votes]: Here is another kind of example where the choice of the generic filter $g$ used to build a generic extension $M[g]$ of a transitive model $M$ is important.  Suppose that we have some set $x$ that we want to be an element of the generic extension $M[g]$.  If $x$ is not already in the ground model $M$, then a mutual genericity argument shows that for any condition $p$ there are lots of generic filters $g$ containing $p$ with $x \notin M[g]$.  So if we want $x \in M[g]$ then we must choose $g$ rather carefully.
For example, if $M$ is a countable mouse with a Woodin cardinal and $x$ is a real, then by Woodin's "genericity iteration" theorem (or a variant due to Neeman) there is an iteration $i: M \to M^*$ of $M$ and an $M^*$-generic filter $g$ such that $x \in M^*[g]$.  This theorem is used all the time in inner model theory, and there is no way to state it using only the forcing language of $M$ or of $M^*$.