User blog comment:P進大好きbot/New Googological Ruler/@comment-31580368-20190629142620/@comment-35470197-20190724014832

> 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.

I do not intend for them to define the same thing. Even if they do not understand a definition of a large number N, they can define N by copying its definition. As I clarified in "Regulations" section, I require that this ruler scales the values of computable functions whose computation rules are precisely described by the authors and whose inputs can be explicitly described as meta-theoretic natural numbers. It is much more difficult to go beyond the least transcendental integer by a system with explicit full rule sets.

I agree that reaching level 25 from level 24 is much easier than reaching level 22 from level 23. But I am quite certain it is also much easier than creating a notation of PTO(Z_2) level. That is why I set it as a "natural" next step in the current googology. The situation will possibly change, when we find a new strategy between them. Then I need to make another ruler. Until the big breakthrough, I think that the nateither.urality of the steps is reasonable. Of course, we can skip levels 22, 23, and 24, because level 25 requires distinct strategies.

> 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.

It might be true. But this is actually one of valid meaningful goals for several googologists, even if they do not reach near up to now.

> 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.

Right. Many googologists just believe hyp cos, because he is one of the greatest active googologists. I am not doubting himself, while I am not seriously believing in the strength or the termination. I can just say "I do not know well about it". I wonder why several googologists talk to me as if the termination is doubtless.

> 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.

It might be true. I just guess that it is more possible that there actually exists such googologists than that there are active googologists who are working on PTO(Z_2).

> Z2 (second order arithmetic) is pretty much the holy grail of ordinal analysis right now.

Right. I know the pure mathematical side. If one find an ordinal notation which can be applicable to Z_2 in order to sclale its PTO, it should be a big news.

> 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.

Ok. If there appears a googologist near the territory, I need a new ruler. At least, my aim is to give a ruler which is applicable to the current googology. The "current" means, say, 5 years. Honestly, I do not expect the appearance of such a googologist within 5 years.

> 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.

I have every considered such regulation three times. However, I could not find the precise description of "explicitly built". For example, we can explicitly write down the full rule set to compute the least transcendental integer. At least, I personally have written full rule sets to compute a large number beyond it, in order to explain how the transcendental integer is not cheating, because I heard several times that the computability was doubtful or unacceptable because no one explicitly write down the full rule sets last year in Japan.

> 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?

Let us start from an elementary case. The termination of f_{ε_0}(n) in not provable under PA under the assumption of its consistency. More precisely, the sentence "forall n, the computation of f_{ε_0}(n) halts" is not probable under PA. On the other hand, for any natural number n (in the meta theory in which we ask the provability under the formalised PA), the sentence "the computation of f_{ε_0}(1*n) halts" in provable under PA, where 1*n denotes the formalisation of n, i.e. the constant term in PA defined by S…S(0) (n S's). This is the pointwise well-definedness.

Finite promise game generates a formula P(n,m) which works as a function in SMAH+, i.e. th sentence "for any n, there uniquely exists an m such that P(n,m)" is provable under SMAH+, according to Friedman. It is not provable under ZFC under the assumption of its consistency, and hence it does not define a function N -> N in ZFC.

The trick is, for any natural number n in the meta theory, the sentence "there uniquely exists an m such that P(1*n,m)" is provable under arithmetic according to Friedman, and hence under ZFC. As a result, the large number defined as the unique solution m of P(2^1000,m) is well-defined under ZFC. An issue is the computability of the function even under SMAH+.

The TI employs the same strategy. The termination of TI (if suitably formalised) is not provable under ZFC, but it is pointwise well-defined. Say, the least transcendental integer TI(2^1000) admits an explicit computation rule, whose halting is provable under ZFC. I myself often use the pointwise well-definedness when I create large numbers, because it can help us go beyond the wall of PTO and eventual domination. The elegance of finite promise game is that it ensures the pointwise well-definedness just using an explicit game instead of proof theory itself.

Since this comment is long, I separate it into two comments.