[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'.