User blog comment:Edwin Shade/The Grand List Of Transfinite Ordinals/@comment-30754445-20171130012046/@comment-30754445-20171130063437

Regarding your question:

It's not a question of "time", but a question of finding an approach that works for you.

I've also been stuck for years with Ordinal Collapsing functions, until I've found some website that gave examples which made everything clear to me. Then, I've progressed very quickly.

Here are a few tips:

1. At first, limit yourself to a single collapsing function Ψ(x) and a single uncountable ordinal Ω. Don't bother with multiple functions (ψ2(x),ψ3...) or multiple cardinalities (Ω2, Ω3...) until you've mastered the first one. This already gets you to the Bachman Howard Ordinal, which is a big leap from the LVO.

2. Forget the formal definitions. They are probably the worst possible way to introduce the topic. Instead, try to get an intuitive grasp of how OCF works. For starters, remember these two simple rules:

(i) Ψ counts the epsilon numbers (IOW: an ordinal x can be written as Ψ(y) for some y if and only if x is an epsilon number)

(ii) You are allowed to use addition, multiplication and exponentiation as well as the Ψ function itself to get out of "being stuck". But you're not allowed to use anything else (in particular: you're not allowed to used Veblen functions).

(iii) Whenever we "run out" of ordinals, we use "Ω".

3. Treat the above as a puzzle or a game. Try to actually "count" up the ordinal ladder with these rules. And by "count" I don't mean just listing the major milestones. Go into enough detail that you're left with absolutely no doubt as to what's going on.

4. Use other notations (epsilon and zeta numbers, Veblen functions, etc) for as long as you can to keep track of where you are. Obviously this cannot be done beyond the LVO = ψ(ΩΩ ), but if you do this systematically then you'll have a good grasp of the OCF concept way before that point.

5. If you have any questions along the way, don't hesitate to ask.

As for the general idea of feeling accomplished: The only competition you're supposed to be in is with yourself. I'm currently stuck only one level above you (I get basic OCF, but anything beyond that is really really vogue to me), but it doesn't matter. What matters is that I'm learning new things and progressing relative to where I was in the past.

Also, keep in mind that all the examples you've given of "Googology Greats" are not the people who invented the largest numbers. Bowers' notation isn't well defined above ε₀, and Saibian's system is limited to well below the SVO. They are renown because they've added their personal touch and created systems that represented their specific leanings (Bowers with his geometrical structures, and Saibian with an extension of a notation he invented in childhood).

As for the fundamental place you can base everything on: The natural numbers are a good start. Sure, Godel and others have dissected math further, but who cares? There's nothing simpler and more fundamental than 1,2,3,... . As the saying goes, you can count on these numbers (literally).

And a second pillar you can rely on are the ordinals. The nice thing about ordinals is that they are... well... ordered in an unambigious way. They form a really neat structure which you can trust and rely on at all times. Sure, we can only grasp a tiny portion of this structure, but this doesn't make the stuff we do understand any less reliable.

At any rate, if you have this feeling of "a rug being pulled under my feet" then perhaps you've jumped to learning advanced stuff before mastering the basics. I'm aware that many people here do this, but it's still a counterproductive practice. No one is chasing us, so why the hurry? Also, remember that all our work here is nothing when compared to infinity. It's a race we cannot win, so there must be some other point to what we're doing here besides "winning".