Are there any known statements that are ***provably*** independent of $ZF + V=L$? A similar question was asked [here](https://mathoverflow.net/questions/11480/on-statements-independent-of-zfc-vl) but focusing on "interesting" statements and all examples of statements given in that thread are not provably indepedent of $ZF + V=L$, they all raise the consistency strength bar. For example, the claim that "there exists an inaccessible" is independent of $ZF + V=L$, is really just an assumption. Because of Gödel's second incompleteness theorem, we cannot prove this. It is well possible that $ZFC$ proves "there is no inaccessible". The same holds for $Con(ZF)$ or "there is a set model of ZF". Those are assumed to be independent of $ZF + V=L$, but this cannot be proved without large cardinal assumptions.


So my question is: Is there any known (not necessarily "interesting" or "natural") statement $\phi$ and an elementary proof of $Con(ZF) => Con(ZF + V=L + \phi) \wedge Con(ZF + V=L + \neg\phi)$? Or is there at least a metamathematical argument that such statements should exists? (Contrast this with the situation of $ZFC$ and $CH$!)


And if not: Might $ZF + V=L$ be complete in a weak sense: There is no statement provably independent of it?


What is known about this?



On statements provably independent of ZF + V=L
The Incompleteness theorem provides exactly the requested independence. (But did I sense in your question that perhaps you thought otherwise?)


The Goedel Incompleteness theorem says that if T is any consistent theory interpreting elementary arithmetic with a computable list of axioms, then T is incomplete. Goedel provided a statement σ, such as the "I am not provable" statement, which is provably equivalent to Con(T), or if you prefer, the Rosser sentence, "there is no proof of me without a shorter proof of my negation", such that T does not prove σ and T does not prove ¬σ. 


This establishes Con(T) implies Con(T + σ) and Con(T + ¬σ), as you requested. [Edit: one really needs to use the [Rosser sentence](http://en.wikipedia.org/wiki/Rosser%27s_trick) to make the full conclusion here.]


In particular, this applies to the theory T = ZFC+ V=L, since this theory interprets arithmetic and has a computable list of axioms. Thus, this theory, if consistent, is incomplete, using the corresponding sentences σ above. Since it is also known (by another theorem of Goedel) that Con(ZF) is equivalent to Con(ZFC + V=L). This establishes the requrested implication:


* Con(ZF) implies Con(ZFC + V=L + σ) and Con(ZFC + V=L + ¬σ)


The Incompleteness theorem can be proved in a very weak theory, much weaker than ZFC or even PA, and this implication is similarly provable in a very weak theory (PA suffices).


One cannot provably omit the assumption Con(ZF) of the implication, since the conclusion itself implies that assumption. That is, the existence of an independent statement over a theory directly implies the consistency of the theory. So since we cannot prove the consistency of the theory outright, we cannot prove the existence of any independent statements. But in your question, you only asked for relative consistency (as you should to avoid this triviality), and this is precisely the quesstion that the Incompleteness theorem answers.


