User blog comment:Kyodaisuu/English description of Fish numbers/@comment-25418284-20131102173245/@comment-5529393-20131103232150

@Kyodaisuu: It's true that you can take the Rayo function for any formal system and get a faster function by adding the Rayo function as an oracle, but you sound a little like those people who say, "No matter which number you pick, I'll just add 1 to it!" It's true that adding 1 to a number will also get you a bigger number, but we reject that extension because there are more powerful ways to get bigger numbers. Similarly, while you can add your oracle function to any formal system, I bellieve there are stronger extensions, like LittlePeng's idea of going to a higher order set theory. The fact that you can add an oracle function to higher order set theories is as irrelevant as the fact that you can add one to any number.

Also, FOST goes far beyond Goucher's Xi function, so we will have already surpassed Goucher's ordinal.

Besides going to higher order set theories, other ideas for extending FOST include adding truth predicates and adding superclasses. Adding a truth predicate will allow you to define the Rayo function within FOST, so it is at least as strong as adding an oracle function. It might be stronger, but I have to think about it.