User blog comment:MachineGunSuper/The Hypersonic Triangle Array Notation (REAL final notation)/@comment-30754445-20181118010314/@comment-30754445-20181119103559

Your verbal explanation vaguely resembles the way Veblen functions work.

The trouble is, unlike Veblen functions, your notation cannot represent the vast majority of the ordinals involved. I'm not even talking about having the notation well-defined. I'm talking about the ability to write down all the intermediate levels.

And remember: It doesn't really matter how you call your letters. Saying that B is ε₀ does not automatically gives you an ε₀-level notation, unless your recursion actually goes through every single ordinal in that range. Worse: It has to be done in the correct order (that's why these entities are called "ordinals". They represent increasingly complex ways to order things).

This, by the way, is also true for ordinary numbers. If I define:

a0(n)=n^n

a1(n)=a0(a0(...(n)...)) repeated n times

a2(n)=a1(a1(...(n)...)) repeated n times

a3(n)=a2(a2(...(n)...)) repeated n times

a1,000,000(n)=a3(a3(...(n)...)) repeated n times

Then I've only done 4 levels of recursion. Indexing the final function with "1,000,000" does not magically give me the power of a million recursions.

To get a million recursions, you need to create a notation that actually counts to a million, like this:

b0(n)=n^n

bm+1(n)=bm(bm(...(n)...)) repeated n times

The "m+1" bit corresponds to ordinary counting, and by setting m to 1,000,000 you get a million recursions.

Of-course, with finite numbers this is trivially easy. With ordinals it is more diffcult. In fact, it gets harder and harder as the ordinals grow larger. But the principle remains the same: If you want to create (say) a ζ₀-level notation, your notation must actually count all the way up to the ordinal ζ₀. No shortcuts. No skipping numbers/ordinals. Yes, this is difficult, but that's one of the reasons that googology is so exciting.

P.S.

You should not be attempting ζ₀-level notations at this time. Such a notation would be several conceptual levels beyond where you're currently standing. Take it one step at time.