TITLE: On statements independent of ZFC + V=L
QUESTION [16 upvotes]: Let $V=L$ denote the axiom of constructibility. Are there any interesting  examples of set theoretic statements which are independent of $ZFC + V=L$? And how do we construct such independence proofs? The (apparent) difficulty is as follows: Let $\phi$ be independent of $ZFC + V=L$. We want models of $ZFC + V=L + \phi$ and $ZFC + V=L + \neg\phi$. An inner model doesn't work for either one of these since the only inner model of $ZFC + V = L$ is $L$ and whatever $ZFC$ can prove to hold in $L$ is a consequence of $ZFC + V=L$. Forcing models are of no use either, since all of them satisfy $V \neq L$.

REPLY [8 votes]: What I would call the standard source of examples is the series of very nice "finitary" combinatorial statements that Harvey Friedman has been working on. You can see plenty of such statements in his numbered series of FOM posts, and for details of some of the arguments, see for example his nice paper "Finite Functions and the Necessary Use of Large Cardinals", Annals of Math., Vol. 148, No. 3, 1998, pp. 803-893.
The examples Harvey examines are arithmetic. In the "true" model (which is well-founded) they are decided one way, and there are ill-founded (in fact, not $\omega$-) models where they are decided the other way.
It would be highly desirable to have examples of nice arithmetic statements that are not just independent but for which we can produce differing answers in different well-founded models. I don't think anybody has any clue at the moment on how to do such thing. To achieve this would be akin to the invention of forcing.