User blog comment:P進大好きbot/Relation between an OCF and an Ordinal Notation/@comment-30754445-20180810102658/@comment-35470197-20180810215606

@PsiCubed2

> Personally, I don't see the problem with the statement "an OCF is an ordinal notation". > > Sure, it isn't precise, but the intention clear. What people actually mean by it is "an OCF-based ordinal notation".

Exactly, if googologists highly understanding set theory, e.g. you, state "an OCF is an ordinal notation", then I intreprete the satement into the tautology "an OCF-based ordinal notation is an ordinal notation".

I am mentioning to the case where googologists without using set thoery state it. Sorry for the ambiguity. (But I believe that you understood that I did not mention to googologists highly understanding set theory.)

> Nevertheless, it's the most efficient way we currently know to construct large ordinals. The only way that allows us to follow the process explicitly. This is required, for example, if we want to create a googological array notation, because we need to know the "expansion rules" at every step of the way.

Hmm. You mean, my ordinal notations do not work well, right? Although they are useless for analysis because we have no way to write down tables, they yield the largest computable numbers among what I have created. I constructed explicit expansion rules, and actually wrote the formal proof of the termination.

I note that since I used the definition of the PTO of ZFC in the constructions, the limit of my ordinal notations are weaker than it. On the other hand, if what Rathjen wrote is correct, then I guess that one of my ordinal notation actually goes beyond his OCF with weakly Mahlo, because I carefully diagonalised the syntax theoretic estimation of the complexity of formal proofs in presenting mathematical papers. (This is also one reason why I am looking for the hidden source of Rathjen's result.)

If you have doubts, please take a look at my ordinal notation here. Then you find full definitions and formal proofs. (The "desired table of analysis" there is just kind of a joke based on custom in this wiki, but might be helpful to understand the image.)