User blog comment:Kyodaisuu/English description of Fish numbers/@comment-25418284-20131102173245/@comment-24061664-20131102223537

Suppose that you can write Rayo(Rayo(i)) in j characters FOST, where i < j < Rayo(i). That means Rayo(Rayo(i)) < Rayo(j), therefore Rayo(i) < j, which is a contradiction

Therefore, you cannot write Rayo(Rayo(i)) in j characters FOST.

By introducing oracle f as Rayo, it is simple to write f(f(i)). Therefore, introducing oracle formula is much stronger than Rayo function.