**There are statements which are  [independent][1] but not provably independent**

If the independence of a statement is a meta-mathematical theorem, then the existence of statements which are independent but not provably independent is a meta-meta-mathematical theorem.    
See the post:  [Are there statements that are undecidable but not provably undecidable][2]      (*undecidable* is here synonymous of *independent*) and the positive [answer][3] (under ZFC, assuming its consistence).


  [1]: http://en.wikipedia.org/wiki/Independence_%28mathematical_logic%29
  [2]: https://math.stackexchange.com/questions/65248/are-there-statements-that-are-undecidable-but-not-provably-undecidable
  [3]: https://math.stackexchange.com/questions/65248/are-there-statements-that-are-undecidable-but-not-provably-undecidable/65302#65302