User blog comment:P進大好きbot/New Googological Ruler/@comment-31580368-20190629142620/@comment-39541634-20190722160305

>"I defined the ruler so that it is suitable for a googologists to aim at the next level without skipping to study mathematical stuff"

I do not understand this statement. Please elaborate.

>"for each level N, there is expected to be a googologist of level N"

I have two comments about this:

1. We currently don't have any level 23 googologists here. Even level 22 is doubtful. Case in point: We do not have notations that reach these levels. We may have lots of wishful thinking and speculation, but nothing more. So your level 23 is just as beyond us (at the moment) as 2nd order arithmetic.

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?

> But don't you think that if googologists understand large numbers of levels below PTO(Z_2), then they will usually aim at transcendental integers before working on discovering fundamental sequences?

That depends on their goals.

If the only thing you're interested in is "getting the largest computable number I can", then you wouldn't bother with anything beyond level 15-20 and jump directly ahead to the "transcendental integers" trick.

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.

Don't get me wrong: It's a very neat trick, and it is also something that any serious googologist should learn at some point. But it isn't really comparable to any of the other levels on your list. It's not "the highest level" to aspire to. Keeping the analogy of levels and video games: TI's are like a "bonus level" that you unlock at some point. You play it, marvel at how cool it is, and then return to your original slow grind at the exact point you left it.

>>For any theory X, we can define f_PTO(X)(n) as follows:

>Maybe many googologists tried the same thoughts (even if they do not know the definition of "proofs"). I have already argued on essentially the same formulation with Scorcher007 above.

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.

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?