I am probably confused about this, but seems to me like you made a switch between “a system cannot prove its own consistency” and “a system could prove its own consistency, but we shouldn’t trust it anyway for outside-of-argument reasons”.
You are correct. Maybe I should have made that clearer.
My interpretation of the impossibility is that the formal system is self-aware enough to recognize that no one would believe it anyway (it can make a model of itself, and recognizes that it wouldn’t even believe it if it claimed to be consistent).
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”.
I am probably confused about this, but seems to me like you made a switch between “a system cannot prove its own consistency” and “a system could prove its own consistency, but we shouldn’t trust it anyway for outside-of-argument reasons”.
You are correct. Maybe I should have made that clearer.
My interpretation of the impossibility is that the formal system is self-aware enough to recognize that no one would believe it anyway (it can make a model of itself, and recognizes that it wouldn’t even believe it if it claimed to be consistent).
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.