[tux3]

It's simpler than this still. If it runs forever (likely), then you will never be able to say anything about ZFC.

If you see it halt, ZFC is inconsistent. If you never see it halt, you CAN'T conclude anything.

But we could already do that under Gödel incompleteness, so there's nothing unusual there!

If you write down random proofs on paper and find a correct proof that leads to contradiction, you've proved ZFC inconsistent, without using BB. If you keep trying forever and never find one, you'll never be able to conclude anything at any point, just like with watching the machine run

[l33t7332273]

> If it runs forever (likely), then you will never be able to say anything about ZFC.

But if you run it for BB(754) many steps, you will know.

[tux3]

Yep. But I think it's easy to show that this is circular, since you can't know BB(754) without knowing whether it runs forever.

And you can't prove that it'll run forever without seeing it go past BB(754) and still keep going

BB(754) is X if ZFC is consistent, Y otherwise

Since you can't prove that ZFC is consistent (only disprove), you can't know BB(754), which is the thing we were trying to use to determine whether ZFC is consistent in the first place!

The definition doesn't make it obvious, but this is just the same as plain Gödel incompleteness, we can't get any extra info about ZFC even in principle (unless we happen to see it halt, by chance)