(This was first posted to math.stackexchange but had no answers there after several days):
Let ${\mathbb R}$ be the set of real numbers in whatever is your favorite model of $ZFC$. Then (by Levy collapse) there is some larger model in which ${\mathbb R}$ is countable.
I wonder whether there are any mathematically interesting facts about ${\mathbb R}$ that are, on the one hand, entirely internal to the original model, but on the other hand, best proven (or perhaps best discovered) by reference to the countability in the larger model.
In other words, I am looking for arguments of the following form: "Choose an enumeration of ${\mathbb R}$ in some larger model $M'$. Then ….. and therefore $X$'', where $X$ is a statement about the real numbers that would both make sense and be interesting to a person who had never heard of model theory. Is there a reason to believe that no such arguments are likely to exist?
Best Answer
In the theory of Borel equivalence relations, one can define something called a pinned equivalence relation. Roughly speaking, a Borel equivalence relation $E$ on a Polish space $X$ is unpinned if there is a forcing which adds $E$-classes consisting only of new elements of $X$. (For a more precise definition, see http://users.math.cas.cz/~zapletal/f2note.pdf. Kanovei's book also has a good discussion, though I don't have my copy on me currently to give a more specific reference than that.) So a Borel equivalence relation is pinned if any new elements of $X$ added by forcing are $E$-equivalent to old elements of $X$ (again, roughly speaking).
Many Borel equivalence relations are pinned. For example, every countable Borel equivalence relation is pinned. Also notably, the relation $E_{l^\infty}$ on $\mathbb{R}^\mathbb{N}$, which makes two sequences of reals equivalent if their $l^\infty$ distance is finite, is pinned. It's also not too hard to show that being pinned is closed downward under Borel reducibility, i.e. if $E$ Borel reduces to $F$ and $F$ is pinned, then $E$ is pinned.
On the other hand, let $E_{ctble}$ be the equivalence relation on $\mathbb{R}^\mathbb{N}$ given by $(x_n) E_{ctble} (y_n)$ iff $(x_n)$ and $(y_n)$ enumerate the same set of reals. One can see that $E_{ctble}$ is unpinned by doing a collapse forcing. If you collapse the continuum to be countable, then you get elements of $\mathbb{R}^\mathbb{N}$ in the new model which enumerate all of the old reals. Clearly these sequences are not $E_{ctble}$-equivalent to any of the old elements of $\mathbb{R}^\mathbb{N}$!
This means that $E_{ctble}$ does not Borel reduce to $E_{l^\infty}$. I would say that this is a fact about (equivalence relations on) $\mathbb{R}^\mathbb{N}$ which is completely internal to the model, but as far as I know, right now the only way we know to prove it is through this forcing argument.