User blog comment:P進大好きbot/What does a computable large number mean?/@comment-4224897-20180610135217/@comment-35470197-20180613122057

Right. It was obvious. Sorry.

But I think that the assumption of such soundness is essentially necessary in the argument, and hence it is impossible to cover all (reasonable) formal theory. For example, if you add to \(T\) a constant term symbol \(M\) and an axiom saying "\(M\) is a halting Turing machine", then it can prove the hating of \(M\) but its output is not presentable in the form \(0+1+\cdots +1\).