User blog comment:P進大好きbot/New Googological Ruler/@comment-31580368-20190629142620/@comment-39541634-20190723134802

>For example, when you have already diagonalisation of matrices, then you can learn Jordan normal form. This is a candidate of the next level in my sense. Of course, there are intermediate topics, which can be candidates. On the other hand, elementary divisor theory needs more detailed study on rings and modules, therefore it is not suitable as the next level. I set the next level as the one which requires mathematical stuffs which googologists at the current level can safely learn.

Now I understand. Thanks.

I still don't get how Level 24 fits your "levels of learning" idea. How do you get from level 23 to transcendental integers as "the next natural step"? Any person who mastered the insane technical complexities of level 23, would - in all probability - already knows how to define transcendental integers. And if he didn't, he would be able to learn it in 5 minutes.

> At least, a Japanese googologist who is working on ordinal analysis (in pure mathematics) is trying to understand the corresponding paper. I guess that Hyp cos also learned something related to them, because he created an OCF based on statbility. (I have not talked to Hyp cos so many times, and hence I might be wrong.

See my comment on Deedlit vs Rathjen on my other post.

Yes, people are working on these things. But they are mostly doing it in a vacuum and without any peer review, so the odds of their work actually amounting to a "level 23" are practically nil.

Take Hyp cos' SAN for example. Has anybody checked his work at the higher levels? No. CAN anybody check his work at the higher levels? No. Did he ever try to give anything more than nebolous justifications to his analysis? No.

And does doing googology in this way EVER ended well? BEAF, BM1, BM2, UNOCF and Deedlit's notation prove otherwise.

So I stand by my statement: Given past experience, we can be quite confident that no actual level 23 googology is happening right now. The only difference between the SAN and (say) BMS is the point at which people start making stuff up.

>Uh-huh. I did not know that these [PI12-CA and Z2] are long-terms goals.

Z2 (second order arithmetic) is pretty much the holy grail of ordinal analysis right now. No googologist is actively working on it (yet) because it is vastly beyond our current capabilities. There are no current OCFs that reach anywhere near PTO(Z2), so this is really uncharted territory. But it is a major long-term goal.

You might also want to put a few intermediate steps between ordinary stability and PI12-CA.

> Hmm... I have a different point of view. The system of transcendental integers is not somethig like cheating, but is actually a great computable googology. I actually regard studying how to go beyond it by an explicit computation rule as a quite interesting topic.

I wasn't aware that this was a thing.

In that case, I believe the best approach would be to create a new seperate scale of levels for these advances. Just like you have two seperate scales for uncomputables. We could have one computable scale for "explicitly built recursive notations" and another computable scale for other computable things.

Because the two lines of research don't really have much to do with one another. We could have Loader's number as level 1 and TI's at level 2 and so on.

>For example, finite promise game, which I do not know the computability though, is one of the solutions. Its pointwise well-definedness under ZFC set theory is ensured by reducing to the provability in weak arithmetic, although the definition itself does not refer to proof theory.

This sounds interesting. Can you elaborate on this more?

I know that finite promise games are much stronger than ZFC. So I can they be "pointwise well-defined" within ZFC?

And a related question, which I was wondering about for quite some time now:

How come f_wck1(n) is fundamentally stronger than BB(n)? Sounds quite counter-intuitive to me.

>I think that one of the main reason why googologists tend to regard it as a kind of cheating is because they have not seen the explicit computation rules, and hence I sometimes explain computations.

I think whether it is considered "cheating" depends on the specific challanges we set for ourselves.

If our quest is to create the largest possible computable number with an explicit set of computable rules, then TI's and Friedman games are perfectly valid. But in this case, it would make no sense to delve into insanely complicated ordinal notatoins which take years to master.

OTOH if our quest is to BUILD the strongest googological NOTATION from the grounds up (like Bird's notation or BMS or SAN) then TI's and Friedman games are completely irrelevant.

The problem is that the average googologists commonly mixes up these two challanges.

And another thought just occured to me:

Much of the confusion here stems from the unstated assumption that "larger numbers" automatically means "more advanced". This may have been correct in the early days of googology, but it is no longer true.

As a small example, it would be silly to insist on learning about Transcendental integers before learning about Busy Beavers. The former is clearly a more advanced topic (which relies directly on the latter) yet it produces smaller numbers.

As another example, I think the Pair Sequence System is one of the coolest googological inventions ever made. It may not be the most powerful notation ever devised, but it so simple that a child could understand how it works (though he won't understand why it works). I find PSS to be a far more impressive innovation then cryptic-but-powerful array notations like SAN.

In short: Even in googology, size is not everything. This is why my own "Psi Levels" scale does not even try to order things in terms of "difficulty". It just attempts to give a vivid map of the large numbers landscape.

>I refered to his comment "Θ(t) is computable function which assigns  each n∈N the least natural number greater than or equal to the halting times of Turing machines with input 0 whose terminations admit formal proofs …". Isn't it sufficiently specific?

Not really. Scorcher007 made claims that I would not have made, and went into details that I personally find irelevant to the discussion.

>Oh, I am very sorry, but I did not intend it. I said that "even if they do not know the definition of proofs" because you might doubt my statement since you could guess that there are few googologists who know mathematical proofs. I never think that you are such one. Actually, I am respecting your knowledge and experience on mathematics related to googology.

So you already know that having this discussion with me is not going to be a repeat of your earlier experience. :-)

Want to give it a try?