TITLE: What questions are independent from the axiom of constructibility?
QUESTION [5 upvotes]: Wikipedia gives a list of statements true in L which would be true also for set theory if the axiom of constructibility (V=L) holds. However I wonder about the converse: Are there any important open axioms/hypotheses in ZFC that would not be settled by the axiom of constructibility?

REPLY [5 votes]: There is a paper entitled On the necessary use of abstract set theory by Harvey Friedman (Advances in Mathematics, vol.41, no.3, pp.209-280, doi), in which he shows that some combinatorial propositions about Borel functions are independent of $ZFC+V=L$.