User blog comment:P進大好きbot/What does a computable large number mean?/@comment-4224897-20180610135217/@comment-35470197-20180612222951

> to prove that the state of the machine is \(state_k\) in a proof of length roughly \(k\),

I am sorry, but I could not understand what this sentence means. The definition of \(k\) is the step of the machine to halt under the assumption that we have a proof of the halting of it, and hence is not described by the definition itself in the form \(0+1+1+\cdots +1\), right?

How do you use the assumption of the \(\omega\)-consistency? If you apply the \(\omega\)-consistency to the halting problem, you just obtain that there is an \(h\) of the form \(0+1+1\cdots +1\) such that the sentence "the machine does not halt at the \(h\)-th step" is not provable under \(T\). It does not imply that the provability of the halting at \(h\)-th step. I am sorry if I am insane...