$\omega^{CK}_1$ is the supremum of all the recursive ordinals, where an ordinal $\alpha$ is recursive if there is a computable ordering of a subset of the naturals with order type $\alpha$.

For a Turing degree $D$, we will say that an ordinal $\alpha$ is $D$-recursive if there is a $D$-computable ordering of a subset of the naturals with order type $\alpha$. We will also say that the supremum of the $D$-recursive ordinals is $\omega^{CK}_D$.

This has some interesting properties that connects Turing degrees and countable ordinals. For example, for any countable ordinal $\alpha$ there is a Turing degree $D$ such that $\alpha$ is $D$-recursive (simply choose a ordering of the natural numbers with order type $\alpha$, and construct an oracle that computes that ordering). This in particular implies that supremum of the $\omega^{CK}_D$ over all Turing degrees $D$ is $\omega_1$. Additionally, the order type of the $\omega^{CK}_D$ over all Turing degrees $D$ is also $\omega_1$. Also, for each Turing degree $D$, we can construct an ordinal notation for the ordinals $< \omega^{CK}_D$, similar to Kleene's O.

My question is, has this relationship between Turing degrees and countable ordinals been explored before?

degrees- while $\mathcal{O}^X$ makes sense for anyset$X$, I don't see any way to get a notation system which is degree-invariant. @JoelDavidHamkins Do you know if such a thing exists? $\endgroup$isdegree-invariant. There are a couple ways to state this: one is that $X\equiv_TY$ implies $\sup\{\mu_X(n): n\in\mathcal{O}^X\}=\sup\{\mu_Y(n): n\in\mathcal{O}^Y\}$ where $\mathcal{O}^X,\mu_X$ are the usual notation system and valuation map assigned to $X$ a la Kleene and another is that if $X\equiv_TY$ then an ordinal $\alpha$ - identified with the structure $(\alpha; <)$ - has a copy computable in $X$ iff it has a copy computable in $Y$). Lesson: when relativizing concepts, we have to bevery carefulabout uniformity issues. $\endgroup$6more comments