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

A few remarks:

1. That definition you've posted is a nightmare to work with.

2. The question you've asked actually have surprisingly simple answer.

(i) For all intents and purposes sup≺ simply means sup. The difference is a formality whose purpose is not relevant to understanding Veblen functions.

(ii) ":" means "such that..."

(iii)  "∈On" means "is an ordinal".

(iv) "Lim" is the set of all "limit ordinals". These are the ordinals which can't be written as "α+1". Alternatively, these are the ordinals which have fundamental sequences. Still alternatively, these are the ordinals which can be written as some multiple of ω.

(v) "≻" - the usual meaning of "larger than".

3. There are two "levels" of multivariable phi. The one with a finite number of arguments, and one where the number of arguments can be any ordinal. I strongly recommend you learn the first part seperately. In fact, I strongly recommend that you limit yourself to φ(a,b,c).

4. The usual way we write Veblen functions is quite different than the one given in you're definition:

(i) We don't use subscripts.

(ii) We write everything in reverse order.

So what we call φ(a,b,c) here, would be written as φ(c0,b1,a 2 )

(and if we want to skip, we just insert zeros: φ(c0,b1,a 3 ) = φ(a,0,b,c)