TITLE: When can we reach a real by forcing?
QUESTION [7 upvotes]: I'm sure this is well-known, but: suppose I have a non-constructible real $r\in V-L$. Under what conditions is there a poset $\mathbb{P}\in L$ and a $G$ which is $\mathbb{P}$-generic over $L$, such that $r\in L[G]$? (Note: $G$ is not necessarily a real.)
Obviously it is consistent (e.g., with $V=L$) that the answer is "always;" for a non-trivial example of this, the result holds if $V$ is gotten from $L$ by adding a minimal degree of constructibility. It seems unlikely that the answer would be "always" in general, but I can't come up with a counterexample.

REPLY [5 votes]: See "When is a given real generic over L?" by Fabiana Castiblanco and myself for an answer to this question:  https://ivv5hpp.uni-muenster.de/u/rds/fabiana_ralf.pdf