User blog comment:VoidSansXD/Ordinal notation/@comment-35434595-20190217190037/@comment-30754445-20190219002032

And what's the 75th symbol? Do you have an infinite list of symbols at your disposal by which to define Δn for all n?

As for this:

"You don't need LITERALLY EVERY SINGLE ORDINAL in order for it to be an ordinal notation."

You do, if you want your notation to be of any use for generating large numbers.

Ordinals are nothing more than a sophisticated form of counting. In googology they are used to keep track of the levels of recursions we are doing.

Just like ordinary counting, if we skip numbers we'll get the wrong result. If I define this:

A1(n) = nn

A2(n) = A1n(n)

A3(n) = A2n(n)

A1,000,000(n) = A3n(n)

Then I haven't magically created a level-1,000,000 number. The definition above has 4 levels of recursion, and calling the last level "1,000,000" doesn't change that.

And the same is true for ordinals. If I have a notation that goes like this:

0,1,2,3,4,..., Δ0, Δ1, Δ2, Δ3, ... up to Δn

Then my notation can count up to w*2, with the symbol "Δn" being equivalent to w+n.

If I then decide to write:

Δ0=w

Δ1=e0

Δ2=z0

It doesn't magically increase the power of my notation to z0. I've merely changed the names of the levels which remain the same. You can't use such a notation to create a z0-level number, unless you fill in all the gaps.