User blog comment:Deedlit11/Extending the fast-growing hierarchy to nonrecursive ordinals/@comment-25418284-20130325203802/@comment-1605058-20130330092508

Yes, you are right. \(\omega^{\omega_1^{CK}+1}\) isn't sum of such ordinals. But I think if a given ordinal is Xi-recursive, all smaller ordinals are too, so this isn't big problem.

If we can construct all trees corresponding to \(\omega_{\alpha}^{CK}\), then, asserting there is no paradox, we can apply Ω operator to them. This is oracle for such trees, so it has level   \(\omega_{\alpha+1}^{CK}\). With a bit of work we can then construct rank \(\omega_{\alpha+1}^{CK}\) Busy Beaver function.

Important part is that \alpha is already proven to be Xi-recursive. If we wanted to do this for \(\Phi (1,0)\) we need to prove its index to be, but index of (\Phi (1,0)\) is (\Phi (1,0)\) itself. Maybe we can avoid this circular definition and prove that it is Xi-recursive, but I don't think it'll happen