[cpa]

Mathematicians say that formula F is true in system S iff one can find a proof of F in the context S. So truthness is always considered with respect to some system. Thus, there are no formula which is "true" but that one can't prove, by definition. (That's the kind of smartassery mathematicians like.)

But the Godel's incompleteness theorem states that for all system that includes (Peano's) arithmetic, one can find a formula in this system that has cannot be proved and whose contrary cannot be proved either. And to answer your question, Godel's proof explicitly shows such a formula.

[JadeNB]

Words in mathematics mean what they are defined to mean, so there's no point arguing about the correctness of definitions. Yours is essentially the constructivist perspective on truth; but it is not the only one. It will declare that certain statements are neither true nor false, which some find distasteful.

A more model-theoretic approach says that a formula is true in a theory (not a system, although the word choice doesn't matter) if it is true in all models of that theory—a statement that can (in principle) be concretely verified simply by testing in each such model. Then it may (and will) be true that a statement is true in "all possible worlds", but that there is no way of proving it from the rules that constrain 'possibility'.