According to Godel result, neither ZFC nor other particular theory is strong enough to resolve all questions about, say, Diophantine equations. But maybe we can hope that a sequence of theories will help? It is known that ZFC-1 theory (ZFC + Cons(ZFC) ) is much stronger than ZFC, in sense that now there are theorems with extremely shorter proofs, and many new theorem are now decidable. If we continue this to ZFC-2, .., ZFC-n, ... then ZFC-w which is union of all, ZFC-(w+1) and so on, we can continue to extremely large sets of theories, about all of them we have no doubts, and maybe now for every natural Diophantine equation we can choose a theory witch resolve it? Moreover, if we would be able to imagine non-enumerable set of such theories, may be we could hope that for EVERY Diophantine equation has a corresponding theory from this set in which it can be resolved? Or this is trivially incorrect “conjecture”? It seems that this does not contradict to Godel Theorem, which consider one theory, not a sequence of theories.
Another way of thinking about the same idea is to take only axioms from ZFC but add a new derivation rule, which would say that "from any set of axioms A it follows that A is consistent". With this derivation rule we would derive Cons(ZFC) in one step! So, for some reasons (by the way, I do not understand why) Godel theorem is not applicable here. May we hope that with ZFC extended with such a new derivation rule we can, say, solve all Diophantine equations?