In set theory Borel sets are important, but we don't actually care about the sets. We can about the Borel codes. Namely, the algorithm to generate a given Borel set starting with the basic open sets (intervals with rational end points, or some other canonical basis) and taking countable unions, complements, and so on.

There are many proofs and arguments that look like "A Borel set $X$ is in a certain ideal if and only if it does not contain a real in a certain generic extension". And while that *does* give us some useful information about ground model sets via generic extensions, it is not $X$ itself that contains that generic real, but rather the "re-computation of $X$ from its code".

But what about the *set* $X$? Clearly you cannot add elements to sets with forcing. When does $X$ remain Borel? Or just generally, when does $\Bbb R$ itself stay Borel in its generic extension?

Is there some relatively simple and non-trivial condition on a generic real $c$ for which ground model Borel sets remain Borel? What about when adding uncountably many reals?

Clearly, if we collapse $2^{\aleph_0}$ to $\aleph_0$, then all the sets of reals become countable and thus Borel. But I'm looking for something a bit more robust, such as a simple criterion to check against Cohen, Sacks, Random, etc.