By the same logic, shouldn’t we distrust it even if it proves “2 + 2 = 4”? I mean, it’s sanity is already suspect, and we know that insane systems could have a wrong opinion on all kinds of math problems, so there is no reason to trust this specific proof.
And I feel pretty sure that Gödel didn’t have “also, no formalized system can prove that 2+2=4” in mind.
Related question: what about systems proving other systems to be correct? Is that simply an equivalent of saying “I 100% believe this other guy to be sane (although ignorant about some facts of the world that are known to me, such as his sanity)”? Which for the external observer would simply mean “if A is sane, B is sane, too”.
By the same logic, shouldn’t we distrust it even if it proves “2 + 2 = 4”? I mean, it’s sanity is already suspect, and we know that insane systems could have a wrong opinion on all kinds of math problems, so there is no reason to trust this specific proof.
And I feel pretty sure that Gödel didn’t have “also, no formalized system can prove that 2+2=4” in mind.
Related question: what about systems proving other systems to be correct? Is that simply an equivalent of saying “I 100% believe this other guy to be sane (although ignorant about some facts of the world that are known to me, such as his sanity)”? Which for the external observer would simply mean “if A is sane, B is sane, too”.
But we can totally prove it to be consistent, though, from the outside. Its sanity isn’t necessarily suspect, only its own claim of sanity.
If someone tells you something, you don’t take it at face value, you first verify that the thought process used to generate it was reliable.