Using a forcing extension $V[G]$ to determine properties of $V$.

Question

It is well known that you can use forcing to change the truth value of various sentences ($CH$, $\Diamond$, et cetera). However, often when performing such a construction over a model $V$, the action is generally in $V[G]$ (or some inner model thereof - as when we violate $AC$ in $HOD(x)^{V[G]}$); we want to figure out what goes on in $V[G]$ for the purposes of, say, a relative consistency proof.

There are constructions, however, that also tell us about what goes on in $V$ (possibly in relation to $V[G]$), as when we consider embeddings of the form $j: V \longrightarrow M \subseteq V[G]$. Such embeddings can have quite small critical points (even as small as $\omega_1$), giving us a new perspective on the small uncountable sets of our original model $V$.

My question:

Are there other forcing constructions (i.e. not generic embeddings) that tell us what the structure of our original model $V$ is like with respect to uncountable sets (so not in the trivial sense of showing the consistency of proof-codes in $V_\omega$) and how $V$ relates to $V[G]$? To put it another way, are there other constructions that allow us to see the structure of $V$ more clearly from $V[G]$?

Answer 1

Examples using generic ultrapowers:

(1) (Gititk, Shelah) If there is a total extension of Lebesgue measure, then there is a Sierpinski set (non Lebesgue null set of size $\aleph_1$ each of whose null subsets is countable).

Proof sketch: Force with the null ideal of the total extension and let $j: V \to M \subseteq V[G]$ be the ultrapower embedding with critical point $\kappa$. Using a generic ultrapower argument, Gitik and Shelah have shown that this forcing must add at least $\aleph_1$ random reals. This set of random reals is also in $M$ and is a Sierpinski set. By elementarity, such a set also exists in $V$.

(2) For every set of reals there is a subset of the same outer measure avoiding rational distance.

Proof idea: Assume it fails and get the restriction of the null ideal to some non null set to be isomorphic to a product of random and Cohen forcing. Use generic ultrapowers to argue that this is impossible.

Example using Shoenfield absoluteness:

(3) (Mycielski) If $A$ is a compact subset of plane of positive area, then there exists perfect sets $P, Q$ such that $P$ has positive length and $P \times Q \subseteq A$.

Proof: See the following mathoverflow post.

Answer 2

My favorite family of examples come from the partition calculus. The original proof of the Baumgartner-Hajnal theorem $\omega_1\to (\alpha)^2_n$ for all finite $ n $ and all countable $\alpha $ appealed to the absoluteness of well-foundedness. The homogeneous set was found in an extension where $\mathsf{MA} $ holds, and that means that a certain ground-model tree is illfounded in the extension.

This idea has been used in a variety of similar results: In Schipperus's proof of the topological version of the Baumgartner-Hajnal theorem, in Todorcevic's proof of the extension of the BH result to non-special posets, in several of the partial results towards the conjecture $\omega_1\to (\alpha, n)^3$ for all countable $ \alpha $ and finite $ n $, etc.