User:Triakula/Classes of notations for numbers

Let \(n_\alpha(O)\) to be \(\alpha\)-th notation for \(n\) using some ordering \(O\) of notations. For example, for \(n = 1\) O can be the ordered set \(\{\frac{1}{1}, \frac{2}{2}, \cdots, \frac\frac{n}{n}, \frac{n+1}{n+1}, \frac{n+2}{n+2}\}\). Not sure if the class of all \(O\) is actually a proper class.