User blog comment:P進大好きbot/New Googological Ruler/@comment-31580368-20190629142620/@comment-35470197-20190722233438

> I do not understand this statement. Please elaborate.

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.

> 1. We currently don't have any level 23 googologists here. Even level 22 is doubtful.

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.)

> 2. It stands to reason that one day, we will have googologists that seriously study systems like PI12-CA and Z2. These are reasonable long-terms goals. So why not have a scale that will be able to accommodate this progress in advance?

Uh-huh. I did not know that these are long-terms goals. Actually, I did not hear many about such higher theories here and Japanese googologist community. If it is still commonly regarded as goals by many currently active googologists, it should be included in the ruler.

> But where's the fun in that? Transcendental integers are basically weakened Busy Beavers. They are just a trick which we can use to bypass all the hard work of actually building our recursions, and win the "computable game" on a technicality.

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. For example, finite promise game, which I do not know the computability though, is one of the solution. 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. Formalising (without natural-language-based ambiguity) full rule sets of a system which goes beyond it is actually attracting for me, and several Japanese perhaps agree with it, because I tried to explain how it is actually computable several times. 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've given an explicit specific definition, and I don't see anything in your previous comments to Schorcher007 that is relevant. If you have any objections to my definition, feel free to make them.

I refered to his comment "Θ(t) is computable function which assigns to 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?

> Also, since I took several math courses in college (including set theory and topology), you can bet that I do know what proofs are. So let us keep this kind of sarcasm out of the conversation, alright?

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. Sorry for the ambiguous sentence.