User blog comment:KthulhuHimself/Ordinal Arithmetic Notation (OAN)/@comment-35470197-20191112141154/@comment-27014275-20191117102253

Well, there's a pattern of recursion defined by a countable transfinite ordinal. Fundamentally, things like 4^[2]7 are computable because they are defined using arithmetic (n^[2]m is just n^m). For ordinals larger than 2, we just apply arithmetic in a recursive manner. By defining larger ordinals using only successors and limits of sequences of smaller ordinals defined in this very same system, we allow the notation to rely entirely on arithmetic and layers of recursion.