What is this function on ordinals called?

Question

Define a function $f:\mathbf{ON}\to\mathbf{ON}$ as follows: for each ordinal number $\alpha$ let $$f(\alpha)=\operatorname{ord(}\lbrace \beta<\alpha|\text{ }\beta\textrm{ is a limit ordinal}\rbrace).$$ This function seems useful, as it seems to describe in some sense "how many levels" an ordinal number has, which leads me to think it might have been studied before. Is there some standard name/notation for this function?

Answer 1

An ordinal $\alpha$ is a limit if and only if it can be written as a product $\alpha=\omega \gamma$ for some ordinal $\gamma$. Thus your function $f$ is almost the same thing as division by $\omega$ (ignoring the remainder).

In detail: for any ordinals $\alpha$, $\beta$, with $\beta>0$, there exists a unique pair of ordinals $\gamma$, $\delta$ such that $\alpha=\beta \gamma + \delta$ and $\delta <\beta$. As in the finite case, we call this division by $\beta$, and call $\gamma$ the quotient and $\delta$ the remainder. If the remainder on dividing $\alpha$ by $\omega$ is $0$, then $f(\alpha)$ is the same as the quotient. If the remainder is nonzero, then $f(\alpha)$ is the quotient plus one.

The above assumes that zero is a limit ordinal. If not, then $f(\alpha)$ will be one smaller whenever it is finite and nonzero.