User blog comment:Kyodaisuu/English description of Fish numbers/@comment-25418284-20131102173245/@comment-24061664-20131104002849

@LittlePeng9 @FB100Z

Thank you for your telling me about your works on expressing Xi function and Turing machine in FOST. I will study them and introduce them in Japanese.

@Deedlit11

> 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.

No matter how strong formal system you define, once you define a system, then oracle formula can be added, and hierarchy can be defined. I don't care if it is accepted or rejected. I just define it. Some people may accept and some people may reject it.

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

You completely misunderstand it. As FOST is stronger than Xi function, Xi function is only R[0] or R[1] level in the hierarchy that I defined. It is a far way to go to Goucher's ordinal. Xi function surpassed Goucher's ordinal in FGH hierarchy, but Rayo hierarchy that I defined is much stronger than FGH hierarchy.

> 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.

Don't mix up 2 things, (1) to make stronger expression and (2) to introduce oracle function.

You may be able to find a way to make stronger version of FOST than oracle implemented FOST, but once you define such FOST, oracle can be implemented in that expression. If you just implement Rayo function within FOST, it just doesn't work as I proved. Therefore, adding oracle is a general way for making large numbers, which works in any kind of formal system.