User blog comment:Edwin Shade/An Injection From The Countable Ordinals To The Reals/@comment-30754445-20171219174932/@comment-32876686-20171219195242

@Alemagno12

You are close, but \(Ordname(\omega)\) in fact equals 0.2t3t4t5t6t7t..., because \(\upsilon (\omega ,1)=2\), instead of one, (as \(\omega\) must reduce to the first element of it's fundamental sequence, 1, which then reduces to it's predecessor, 0). Therefore the first sequence of digits to be concatenated after the decimal point is '2' instead of '1'.

@PsiCubed2

Alemagno12 is correct in that \(Ordname(\alpha)=0.\alpha t\alpha t\alpha t...\) where \(\alpha<\omega\), and LittlePeng9 correctly explains how to evaluate \(\omega+1\). The reduction-length of an ordinal is how long it takes to reduce it to 0 for a given n, and therefore is a finite number. The \(Ordname(\alpha)\) function is a function which creates a sequence of digits which when concatenated happens to be a real number. You are right in saying that very few real numbers will be assigned an ordinal, but this is just a proof-of-concept.

There is the issue of whether distinct ordinals will always produce different values when plugged into the Ordinal Naming function, which I have not proven, but intuition tells me it's probably true.