Consider the indexes of all ordinary programs generating functions from $\mathbb{N}^2$ to $\{0,1\}$. If we let $W$ be the set of exactly of all those indexes $e$ such that $\phi_e$ computes a total function that represents a well-order relation (on $\mathbb{N}$) for a recursive ordinal $\geq \omega$ then I think $W$ is not hyperarithmetic. That's because I have read it multiple times on various points.

Now consider the result that any set generated by an OTM (or, I think, reasonably similar model of ordinal comp.) in less than $\omega_{CK}$ time must be hyperarithmetic. Hopefully I am not making a mistake here, but it seems(?) that this result could be used to show that $W$ is not hyperarithmetic (given the assumption that $\omega_{CK}$ is admissible). 

I seems that there is a relatively easy construction that shows that if $W$ was hyperarithmetic then $\omega_{CK}$ wouldn't be admissible. Hence we can conclude that $W$ is not hyperarithmetic. I have a question here that has been bothering a bit for a while, ever since few years ago that I noticed this point.

**(Q1)** The first concern is of course of "circularity". What I mean is that I have no idea what showing $\omega_{CK}$ as admissible really entails in terms of set theory involved. For example, perhaps(?) showing $\omega_{CK}$ as admissible already necessarily uses $W$ as non-hyperarithmetic as a lemma, which would make the above construction perhaps useful as an aid but void as a result. Since I don't know much in the way of detail here I thought it would be reasonable to ask.


**(Q2)** Secondly also I am wondering about one other point. We can use $<\omega_{CK}$-time sets as hyperarithmetic to also show for example the result mentioned by Andreas Blass here  https://mathoverflow.net/questions/278045/mapping-between-notations (see the comments below the answer). In one sentence, somewhat informally, the result is that the "mapping" between any two recursive well-orders must be hyperarithmetic. 

Once again I am wondering whether there is any "circularity" in using OTM to show such a result or not? I am thinking not but I think it is better to be ask, since it is still just a guess on my part.