User blog:PsiCubed2/A smoother ordinal scale up to ω^ω

Let \(f\)  be a function from \(\{\alpha|\alpha<\omega^\omega\}\) into the rational numbers as follows:

1. \((\omega^n \cdot k) = n+1-({{2} \over {3}})^{k-1}\) 2. If \(\alpha=\omega^n \cdot k\) and \(\beta<\omega^n\) then: \(f(\alpha+1+\beta) = f(\alpha)+{{f(\beta)+1} \over {3(n+1)}} \cdot  ({{2} \over {3}})^k\)

This results in a relatively smooth scale of ordinals up to \(\omega^\omega\), with the rational number \(n\) representing \(\omega^n\). Later, I'll also show how this scale can be used to create a continuous \(\omega^\omega\)-level function that's smoother than anything we've ever seen before (and in particular - far smoother than the \(P\)-function) in Letter Notation.