User blog comment:PsiCubed2/For Newbies (and Veterans too): The Great Scale of Googology/@comment-24920136-20171219165253/@comment-30754445-20171221075452

Bold textChrono, do you think it can be done better? If so, I'm open to suggestions.

The whole point of this scale is to help beginners make some sense in the googological world. Can your points be addressed without sacrificing the main goal of the project?

I will preface this reply by emphasising that it isn't some kind of "rebuttal" to the points you've raised. I'm aware that there's room for improvement, and this is why I've asked for feedback. I'm simply posting my thoughts about the points you've raised:

1. It's an unpleasant read:

Then please tell me how you think I could improve the presentation.

How can I make it less unpleasant?

I'll discuss your content-related objections in a moment, but first things first: If the experience of reading the post was unpleasant, then this gotta be fixed before everything else.

2. Arbitrariness:

I honestly think that any such scale would have to be arbitrary. After all, it is theoretically impossible to create a universal "catch-all" function that captures all the levels of googology.

The only question is whether we'll use human choice throughout the entire scale (like here), or break it into arbitrary pieces and define each piece by a formula (like Letter Notation, or the Mashimo Scale).

I don't find the latter approach to be any less "arbitrary" than the former. And given years of complete failure by many people (including myself) to create a smooth and easy-to-use scale with that approach, I've decided it isn't worth the trouble.

At any rate, the arbitrary choices I've made here reflect an attempt to make the scale as easy to use and as easy to understand as possible (which is why - for example, round numbers correspond to significant googological milestones).

And again: If you or anyone else have suggestions about how to make better choices (given the goal at hand), I would very much welcome your input.

3. Guesswork:

There's no "guesswork" in here at all. Any uncertainty stems from the fact that I haven't yet decided which levels correspond to which numbers beyond level 100. Some of it reflects my own ignorance (especially at the higher levels), and it is precisely my aversion of doing "guesswork" than lead me to put those question marks in there. I have absolutely no idea, for example, how far Deedlit's I,M,K system is from 2nd order arithmetic, or how far is nth order arithmetic (Loader's number) from ZFC.

So for me to set the levels of these things in stone would be silly, until an expert on this topics comes here to sort them out ...or until I learn enough to do it myself.

4. Inconsistent Scaling: Here I think you're actually mistaken. Going (say) from ε₀ to ε₀+1 is a much bigger jump than you think. This will be clearer if we divide that portion of the scale further (this is just for sake of example, rather than an actual definition of sublevels): Do you still think the above scale is inconsistent? Suddenly the "small" jump from ε₀ to ε₀+1 doesn't seem so small any more, does it?

See, this was done on purpose, because most people (including some who are actually well-versed in googology) tend to forget this quirk of the googological world: The number scale behaves differently than the ordinal scale, even though there's a connection between the two.

Letter Notation, by the way, does not address this fact properly. This is why, in our MNiB game on XKCD, we spent 95% of our time writing things like "P2.00054". It may technically be a continuous notation, but is so unsmooth and uneven that it isn't even funny (for example, [ε₀]9 ~ R1.954, [ε₀]10 = R2.000 and [ε₀]11 ~ R2.0000000000001)

This is exactly what I was trying to fix with my current scale.

5. Some of the numbers are not well defined:

Agreed.

This, again, reflects my own lack of knowledge. I'm a complete noob when it comes to proof theory, and I have absolutely no idea how to define things like fZ 2 (n) myself.

But this doesn't change the fact that "2nd order arithmetic" is a pretty clear cut-off point. Basically, any function which can be proven to be total in a reasonable amount of steps in Z2, will yield a number below level 250 for any reasonable input.

I'm sure an expert on this topic would be able to properly define fZ 2 (n) by setting the limit of "reasonable" to n (or some relatively modest function of n. 2^^n seems to be a popular choice). And if we give 10 different experts this task, I'm quite confident that the difference between their functions won't change the value of "level 250" by much.

Of-course, once we start filling out the levels on either side (what's 240? 249? 251? 260?), then exact definitions would be important. There's a reason why I haven't yet placed things like "Z2+1" (I'm so clueless at these levels, that I don't even know how sensitive the placement of such ordinals would be to the specific definition of fZ 2 we pick )

5. Missing alot of uncomputables between BB(n) and Rayo(n) (i'm talking about things like oracle BB's, FISH numbers etc..):

Eventually the scale will include them.

The reason it currently doesn't, is that I have absolutely no idea how far (conceptually) Rayo is from Oracle BB's. If we set 400 to BB and 450 to Rayo is 450, where should we put Oracle BB's?

The current picture I have in my mind is:

'''BB → computable recursions on BB's → ? → oracles → ? → Rayo.'''

I have absolutely no idea how big or small these '?' gaps are, so trying to create sensible subdivisions with this kind of limited information would be laughable (I've did put Xi in there, but notice the question mark. I have no idea if 420 is sensible or not).

6. "50 levels don't to the BB/Rayo difference justice":

Rayo is positioned at the halfway point between BB(1010) and infinity. That's not respectable enough?

But perhaps you're right that we need more than 500 levels. Perhaps we should put Rayo at 500 or 600 or whatever, and infinity at 1000.

As I've already stated, I'm open to suggestions for imporvements, as long as they align with the pedagogical goal I've set here.