When using axiom of choice in proofs, people often say that this is non-constructive because AC gives us only proofs of existence, without giving explicit example. However, because in $L$ AC holds, we have that for every existence proof using choice there is an explicit example for that in L. So AC is non-constructive, but it sort of is constructible.

I have recently wondered if we could make up some stronger choice principle, which would allow us to make up sets so complex that they aren't constructible, that is, which could let us prove the existence of non-constructible sets, thus making $V=L$ false.