User blog comment:KthulhuHimself/Ordinal Arithmetic Notation (OAN)/@comment-35470197-20191112141154/@comment-35470197-20191117100514

> Didn't I say that for 0, 1 and 2 we treat ordinal arithmetic as the regular definitions of addition, multiplication and exponentiation respectively?

It is irrelevant.

> What do you mean by encoding it in arithmetic if not that?

First of all, do you understand that arithmetic does not have ability to deal with ordinals? Using ordinals and set theoretic operations such as limits works in set theory, but not in arithmetic. Or don't you know any example of arithmetic? For example, do you know PA? If you know it, then could you understand that transfinite ordinals are not terms in PA?

> Or are you referring to the way finite numbers are derived from the ordinals?

No. But I guess that you misunderstand the computability of FGH. It is easy to see that FGH (a map α×N→N for a fixed ordinal equipped witha system of fudamental sequences) is not computable unless you specify a way to encode α into arithmetic such as PA. by the definition of computability.