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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.

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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.

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