[tybug]

You're right, I misstated this - but I don't think this is fatal. The other sibling commenters pointed out the real issue with my thinking.

The argument goes the same even though I misspoke here. If the machine {halts, runs forever} then ZFC is consistent. But this is a contradiction; so ZFC must be inconsistent. Tada, I have an inconsistency proof!

That was the implied next step which made me think my logic was clearly incorrect (which, it was).

[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)

[bmacho]

> You're right, I misstated this - but I don't think this is fatal.

It is crucial at this types of results, when you search for a proof.

There are a lot of things true, better make a table of it, instead of a wall of text:

  - If you observe the machine halting, ZFC is inconsistent. 
  - If the machine hasn't halted yet, you don't know if ZFC is consistent or not.

  - If ZFC is inconsistent, the machine will eventually halt. (You have an upper bound for this, given a contradiction.)
  - If ZFC is consistent, then the machine won't halt ever.

Also it is consistent with ZFC that this machine halts, since it is consistent with ZFC that ZFC has a contradiction. This means that if ZFC happens to be consistent, and you work in ZFC+contradiction, then you will know that your machine will eventually halt, yet it won't halt ever.