User blog comment:MilkyWay90/Help with understanding Veblen array notation/@comment-30754445-20180811202716/@comment-30754445-20180813092856

@Fefjo

You probably just didn't find the right learning method.

Attempting to learn the topic directly from the definitions is  - indeed - very confusing. I too struggled for over a year to understand the basics of OCFs in this way... without success.

Then I found a very comprehensive article about the subject, complete with a walkthrough and examples, and mastered the entire thing (up to the BHO) in less than a week.

Unfortunately, I don't remember were I've found the article. But if I can find the time, I'll write my own introduction to Madore's ψ in the near future.

At any rate, the subject is notorious for looking way more complicated than it really is. The two reasons it seems difficult are:

(1) The proper definitions look intemidating when written in mathematical form (if I translated Deedlit's BHO level post from "symbolese" to ordinary English, it would have looked a million times simpler).

(2) The entire thing is based on a single concept which is somewhat tricky to wrap your head around. So until you learn how that concept works, nothing will seem to make sense. This gives an illusion of the subject being really difficult to learn, because you just assume that if "nothing makes sense" to you then you'll have to learn two dozen new things to make heads or tails of everything.

The truth, though, is exactly the opposite:

Once you pass that single hurdle (which is - indeed - kinda tricky), everything else would click into place beautifully. The rest of the learning process is just getting used to the new notation and the new idea.

This cannot be said for extended Veblen functions, where you need to memorize quite a few rules and special cases and other irksome things. You're right that it is easier to start learning them, but I don't agree that it is easier to master them.

@Syst3ems

Technically, such a notation could get to the BHO... but only if it incooperates OCF-like principles.

And actually, defining φ(1,0,α) as φ_Ω(α)  (nor sure where Fefjo's "+1" came from) is already doing what OCFs are doing: Using Ω as a wildcard to "fill in the blanks" whenever we run out of ordinals. That's precisely the "big idea" behind OCFs! So to be honest, I'm not sure why Fefjo is having problems with them.