$\omega^{CK}\_1$ is the supremum of all the [recursive ordinals](https://en.wikipedia.org/wiki/Recursive_ordinal), 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](http://if%20there%20is%20a%20computable%20ordering%20of%20a%20subset%20of%20the%20naturals%20with%20order%20type%20$%5Calpha$.) $D$, 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 [$\omega\_1$](https://en.wikipedia.org/wiki/First_uncountable_ordinal). 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](https://en.wikipedia.org/wiki/Kleene%27s_O).


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



Connection between countable ordinals and Turing degrees
The ordinals of the form $\omega\_D^{CK}$, as you denote it, are exactly the [countable admissible ordinals](https://en.wikipedia.org/wiki/Admissible_ordinal), and these ordinals are intensely studied in the context of admissible set theory and fine structure theory. 


