User blog comment:MachineGunSuper/Exploding Fast Growing Hierarchy/@comment-30754445-20171223190206

Well, it looks good up to ω×2. As for the growth rate, it isn't that different than the ordinary FGH, but that's fine. The FGH itself is really really really strong (there's really no limit to as high it can go, because we never run out of ordinals). Generally, it is far more efficient to extend your hierarchy to a higher ordinal, then it is to try and tweak the FGH itself.

At any rate, you're good to go up to ω×2.

But after that, you're beginning to skip steps... which is strange, because you've correctly listed how some of the bigger ordinals work.

You correctly wrote that ω×3 is the limit of ω×2+1, ω×2+2, ω×2+3,... But then completely disregarded this fact when you defined fω×3(n) as a simple repetition of fω×2.

This means that your "₰ω×3" is only level ω×2+1.

You also haven't actually defined all the functions you're using. For example:

₰ω^5(5) = ?

I could guess from your examples that this should be:

₰ω^4(₰ω^4(...(n)...))

But that's just by analogy. Nowhere on that page have you actually given the instructions to evaluate such expression.

Now, if we fill in all these blanks, then it looks like your ₰ω^ω(n) would be at level ω×3 (which - by the way - would answer my 1st challange to you on the other thread, once you fill in all the gaps). Your ₰ε 0 (n) would be at level ω×4, and after that... well, after that it gets fuzzy, because I can no longer even guess what your intentions where.

This is actually pretty good for a second attempt. An ω×4-level number is nothing to sneeze it. The biggest things to improve for next time are:

(1) Be more careful when you define your steps. There should be absolutely no room for guessing regarding what each stage of your hierarchy means.

(2) You might want to create a completely original notation, rather than something based on ordinals. You should be proud of your ability to create ω×4-level number (which is something which was completely beyond your abilities just one week ago), rather than be disappointed that your "₰ε 0 (n)" doesn't get anywhere close to the actual ε0-level.

(there's a reason I've put ε0 at level 50 on my scale. It is much more difficult to get there than you think).