User blog comment:Edwin Shade/My Googological Goals For The Year Of 2018/@comment-32697988-20180101223205/@comment-30754445-20180104155734

Technically, (0,0)(0,0)(1,0) corresponds to the form "1+ω".

In ordinal arithmetics, of-course, 1+ω = ω. But the fundamental sequences are different:

ω = lim (1,2,3,...)

1+ω = lim (1+1,1+2,1+3,...) = lim (2,3,4,...)

So these two forms give rise to two different functions:

Hω(n) = Hn(n) = 2n

H1+ω(n) = Hn+1(n) = 2n+1

The latter is what you've found.

At any rate, it is costumary to never use pair sequences of this sort. We can define a "regular pair-sequence" to be one that can be arrived by the following process:

(1) We start with (0,0)(1,1)(2,2)...(n,n)[k] for some natural numbers n and k.

(2) At each step we expand the string once and - if we wish - change the value of k afterwords.

(3) We may repeat step (2) as many times as we like.

And if we limit ourselves to regular pair-sequence, then each pair-sequence indeed corresponds to a single ordinal.

What's more, by limiting ourselves to regular pair-sequences, we get a very elegant result: if we list the pair-sequences in the order of their corresponding ordinals, they will automatically be ordered in a lexicographic order.

So if you see these two regular sequences:

(0,0)(1,0)(2,0) and (0,0)(1,1)(0,0)(1,0)

Then you can tell at a can tell at a glance that the second one is bigger than the first (because (1,1) comes after (1,0))

(If you're curious, (0,0)(1,0)(2,0) = ωω and (0,0)(1,1)(0,0)(1,0) = ε₀+ω)