User blog comment:P進大好きbot/What does a computable large number mean?/@comment-4224897-20180610135217/@comment-27513631-20180612200337

I might have forgotten a subtlety, so let me do a sanity check:

Suppose we have proved that a Turing machine halts, and that our theory is \(\omega\)-consistent*. Then it halts in \(k\) steps, for some natural number \(k\). Thus, we can use the theory to prove that the state of the machine is \(state_k\) in a proof of length roughly \(k\), and if this state happens to be halted, we've proven in the theory that the machine halts.

* I want to cover all ground - this may be too strong or unnecessary.