User blog comment:Edwin Shade/My Googological Goals For The Year Of 2018/@comment-32697988-20180101223205/@comment-32876686-20180104153531

Your definition was very clear and I could understand it, but I have a question, which is about how you can relate the pair sequence to an ordinal in the Hardy hierarchy.

Here are my results for evaluating \((0,0)(0,0)(1,0)[n]\).

\((0,0)(0,0)(1,0)[n]\)

\(=(0,0)X[n]\), where X denotes n copies of \((0,0)\),

\(=2n+1\).

There is a problem though, because if \(\omega\) corresponds to the growth rate \(2n\) and \(\omega+1\) corresponds to \(2n+2\), then how are we to assign an ordinal to a growth rate of \(2n+1\), which is just in-between the two whole ordinals ?