User blog comment:PsiCubed2/Help Me Understand Ordinals Beyond the BHO/@comment-1605058-20170220180917/@comment-30754445-20170307010000

LittlePeng, I want to get an actual "feel" of how the ordinals work. This means:

1. Being able to intuitively "count" with the system. By "counting" I mean writing a progression of ordinals with arbitrarly-sized skips and always knowing "what comes next".

2. Being able to give - at a glance - a fundamental sequence for any ordinal in the system.

And I want to be able to do both without needing to "peek" at any references.

Presently, I can do both up to the BHO and a little higher (up to around ψ₁(ψ₂(Ω₂)).

As for the actual definitions, I understand the symbols but unable to see the larger picture of how it all comes together. One of the problems is that there are many variaitons and they aren't exactly identical. I've actually dug up a few articles (by Rathjen and others) and each one has a different definition for the ψ functions. And the same is true on this wiki. Even in Deedlit's own article, the ψ in each part seems to be a completely different function (rather than an actual extension). In some parts he is allowing veblen functions and in others he doesn't. In the later parts he allows the operation β →Ωᵦ which is bound to change the values of the lower levels as well.

So I guess, before I even try to actually understand the ordinals, I need some "standard" way to write them. Perhaps I should make my own notation, based on the same principles, and ask for comparisions?

And as I'm writing this, I've thought of another very speciific question:

How sensitive is the rough growth rate of the collapsing function to the details of the definition? IOW if I write my own version of the psi functions, could "my" ψ(ψᵢ(0)) turn out to be bigger or smaller than Rathjen's ψ(ψᵢ(0))?