User blog comment:P進大好きbot/What does a computable large number mean?/@comment-4224897-20180610135217/@comment-27513631-20180613084611

Given any (reasonable) formal theory \(T\), you can show that, after \(k\) steps, the machine is in  simply by showing that it's in state \(state_1\) after \(1\) step, \(state_2\) after \(2\) steps, etc. This can be continued for an arbitrarily long time, and the length of the proof grows at a comparable rate to the number of steps, as we're just proving the state after said number of steps by considering each one in turn.

Now, if a theory \(T\) (which includes, say, something as strong as EFA) is \(\Sigma_1\)-sound (which is implied by \(\omega\)-consistency), then "\(\Sigma_1\)-soundness is equivalent to demanding that whenever \(T\) proves that a Turing machine \(C\) halts, then \(C\) actually halts." (from wikipedia)/ Now, we know that the machine halts after \(k\) steps for some actual natural \(k\), so the theory proves that it halts after \(k\) steps for some numeral \(k\).