# Can it be shown that ZFC has statements which cannot be proven to be independent, but are?

I am familiar with the concept that a statement can be proven indepent such as in the case of the continuum hypothesis where both ZFC+CH and ZFC+(CH is false) are both proven consistent, but I would like to know if any current publications show that ZFC can show the following to be true of a statement T:

[ZFC+T is cconsistent and ZFC+(T is false) is consistent] is independent

(I apologize at how informally the question is posed; I've just started to read the work of Cohen and Godel and have been unable to find any text refering to this problem)

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To nitpick, ZFC+CH is not proven to be consistent. It was proven that if ZFC is consistent then ZFC+CH is consistent (and same for ZFC+~CH). It is unknown whether ZFC is consistent or not, and by Godel's incompleteness we will probably never know as well. – Asaf Karagila Jul 17 '11 at 18:10

On the other hand if ZFC + T is consistent, then it is also impossible for ZFC to prove that "ZFC+ T is inconsistent". Because ZFC is $\omega$-consistent, if ZFC proves some theory is inconsistent then the theory really is inconsistent.
Of course; everyone who works with ZFC assumes it is consistent (in fact as I point out it is $\omega$-consistent). If ZFC is inconsistent then it proves every statement in its language and the question itself is of no interest. At the same time, I refer always to ZFC, not to ZFC + Con(ZFC). – Carl Mummert Jul 17 '11 at 18:50