User blog comment:Kyodaisuu/English description of Fish numbers/@comment-25418284-20131102173245/@comment-24061664-20131103194339

Thank you for summarizing. As Ikosarakt1 wrote in another thread, any hierarchy will meet up to some ordinal. As Rayo hierarchy was defined, it will get to a ceiling point of Goucher's ordinal or somewhere. If you can define another Rayo function for 2nd order set theory, I just define oracle formula in that function in that 2nd order set theory expression, and make another hierarchy. After that, Goucher's ordinal will be reached. They are all same story, but the base expressions are different, 1st order arithmetic, 1st order set theory, or 2nd order set theory.