User blog comment:Plain'N'Simple/A question for proof-theory experts/@comment-35392788-20191029194318/@comment-35470197-20191030222137

@Syst3ms

> I don't really understand your last point

The function \(n \mapsto \alpha[n]\) can be provably to total with respect to a fixed ordinal notation in many cases. For example, remember BMS v. 2.3. It is not literally an ordinal notaion, but assume that we got a way to realise it as an ordinal notation. Then the [n] operation is just a single expansion, which is obviously total on the set of standard BMSs under many weak arithmetics such as PA. On the other hand, it surpasses ε_0, which is PTO(PA).