I find I frequently have to refer to the set of ordinal notations below some given notation. For instance given a notation $\alpha$ I often need to refer to the set $\lbrace \beta \mid \beta <^{\mathcal{O}}\_s \alpha\rbrace$. Higher Recursion Theory simply writes such sets as $W\_{g(\alpha), s}$ (where $g$ is the appropriate computable function) but this isn't very helpful when writing a paper.


I find that when one is working with constructions occurring at infinite ordinal levels (say in $\alpha$-REA sets) these sets pop up everywhere and without a specific notation for them it is unwieldy to refer to them or specify properties of the enumeration. 


Is there a standard notation for what I'm looking for? If not any suggestions?


How about $\mathcal{O}\_{< \alpha, s}$, i.e., the part of Kleene's O below $\alpha$? Writing out $\mathcal{O}\restriction\_{< \alpha, s}$, while correct, seems confusing as it's not obvious what set the stage subscript refers to.


I want to settle on something so I can put it in the next version of my recursion theory latex package.



Using Ordinal Notations in Computability Theory Is There A Standard Notation For The Notations Below $\alpha$
In my thesis (<https://www.lacl.fr/~benoit.monin/ressources/misc/Thesis_report_benoit_monin_v1.9.pdf>), I use $\mathcal{O}\_{<\alpha}$, $\mathcal{O}\_{\leq\alpha}$, and $\mathcal{O}\_{=\alpha}$, for the set of notations of ordinals respectively strictly smaller than $\alpha$, smaller than or equal to $\alpha$, and equal to $\alpha$.


I think these notations are convenient and make sense.


