It proves, that it is impossible to design any formal system, within which it would be possible to prove every principle/aspect that applied to that same system ( aspects that nevertheless WERE true, and CAN be proven by using a 'bigger hammer'.) In a way, the problem is, that whenever you expand your system to make it 'more complete' (to be able to prove more stuff), you'll invariably introduce new aspects you THEN can't cover. You "can't pull yourself up by your own hair".