# Does $\omega_1^{\text{CK}}$ allow to compute the halting problem of $\alpha$-th-order Turing machines for any $\alpha < \omega_1^{\text{CK}}$?

This page contains the following text (see the section “Higher-order busy beaver functions”):

At least, under any reasonable formulation of the notion of a higher-order Turing machine, well-orderings of ordinal type $$\omega_1^{\text{CK}}$$ are uncomputable even with respect to an $$x$$-th oracle Turing machine for any recursive ordinal $$x$$.

What about the other direction? That is, assuming that $$\alpha$$ is an arbitrarily large ordinal less than $$\omega_1^{\text{CK}}$$ and we know what it means for a Turing machine to have an oracle for a copy of some ordinal, is it necessarily true that for any copy $$T$$ of $$\omega_1^{\text{CK}}$$ the halting problem for the family of $$\alpha$$-th-order Turing machines is computable by some Turing machine that has access to an oracle for $$T$$?

I am inclined to think that the answer is “no, it is not true” (if the answer is “yes, it is true”, I would be surprised by how strong $$\omega_1^{\text{CK}}$$ is), but I cannot find the proof.

For any countable linear order $$\mathcal{L}$$ whatsoever and any noncomputable set $$X$$, there is an isomorphic copy of $$\mathcal{L}$$ with domain $$\mathbb{N}$$ which does not compute $$X$$ (or, if you prefer, whose atomic diagram does not compute $$X$$).
This was proved by Linda Richter (Theorem 3.3 in Degrees of structures). In particular, there are copies of $$\omega_1^{CK}$$ which don't even compute $$\emptyset'$$. See Knight, Degrees coded in jumps of orderings for further results along these lines.
• @user21820 Yes, but this has nothing to do with $\omega_1^{CK}$: any nontrivial (in a particular sense) structure has copies computing arbitrarily complicated reals. In fact, Julia Knight showed (in the last section of the linked paper) a stronger result, that if $\mathcal{A}$ is nontrivial then the set degrees of copies of $\mathcal{A}$ is closed upwards under Turing reducibility. Commented May 9 at 14:40