Wouldn't the 'truth' of a statement be depending upon the axioms of where you are making the statement. Thus something proven is true and if the negation is pr

Wouldn't the 'truth' of a statement be depending upon the axioms of where you are making the statement. Thus something proven is true and if the negation is proven it is false, and thus a system able to prove both is self contradictory and even basic logic no longer applies, thus we lose any real world application. The opposite a system which has true statements we can't prove is still useful, even if there might be some problems we won't ever be able to solve. But a system which can prove both a statement and its negation loses meaning. Or does it? Naïve set theory, despite Russel's paradox, and language in general, despite the local equivalent of "This statement is a lie." is still a useful tool, so maybe the same applies even to more formal systems?