User blog comment:P進大好きbot/Relation between an OCF and an Ordinal Notation/@comment-30754445-20180810102658/@comment-30754445-20180811190646

@P-bot

"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".'

That's not a "tautology". It's an explanation of how the term "OCF" is used here.

It isn't any more of a tautology then saying "an even number is an integer" (even though the word "integer" appears in the definition of even numbers). It's not a tautology because the converse is not true: Not ever ordinal notation is an OCF (in the common usage of the term here).

"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.)"

As far as I can tell, they mean exactly the same thing as I do, when they say "OCF". Just because they lack the experience to define their usage precisely, doesn't mean that they don't understand what they're saying.

Besides, the way I see it, these kinds of things are only a problem if they result in sloppy work and/or in people making erronous deductions.

Can you give an example where the imprecise usage of the term "OCF" results in such a problem? If not, then it's just a semantic difference.

There is one point of confusion, though, that I do think is important:

Not every ordinal notation that looks like an OCF, is indeed an OCF. And unlike the semantic issue we've discussed earlier, this is important.

I don't mind it at all, when use "OCF" to refer to - say - Deedlit's oridnal notation or even UNOCF. What I would mind, is if someone made a jumble of definitions and symbols, and declared he had created an OCF just because of some vague visual similarities.

"Hmm. You mean, my ordinal notations do not work well, right? Although they are useless for analysis because it is very difficult for human beings 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."

There are no "explicit expansion rules" there.

Defining a set of expantion rules formally, and actually listing them explicitly are two different things. This is precisely the difference between what I call a "step-by-step" notation and a "proof-theory based" notation. And it is the former type, that are currently limited to ordinals far below PTO(Z2). And given that your notation relies heavily on godel numberings, it is clear that it is of the latter type.

By the way, your idea reminds me of Kleene's O, which is an actual notation that reaches ω1ck. The catch is that Kleen's O is uncomputable (as are all ordinal notations that reach that level).

And while your notation may be computable (I'm not sure because I haven't understood everything you've written there), it suffers from a similar catch: Writing the explicit rules for it are practically impossible (even though it may be possible in principle).