User blog comment:KthulhuHimself/Ordinal Arithmetic Notation (OAN)/@comment-35470197-20191112141154/@comment-27014275-20191117091013

Not entirely certain, but I'll stop roughly when it becomes difficult to extend in meaningful ways.

And I've defined a well-ordered system for these ordinals through which recursion (which is computable) can be achieved. If you look at the definition, you'll see that it is indeed encoded in arithmetic.