User blog comment:Deedlit11/Extending the fast-growing hierarchy to nonrecursive ordinals/@comment-25418284-20130325203802/@comment-5529393-20130330023258

An ordinal isn't necessarily a sum of admissible and recursive ordinals. Take \(\omega^{\omega_1^{CK} + 1}\), for instance.

I don't follow when you say "By allowing  oracle operator to work on all these ordinals,  \(\omega_{\alpha+1}^{CK}\) is Xi-recursive." Are you suggesting that if you have a sequence of SKIΩ trees corresponding to ordinals which have a supremum of \(\omega_{\alpha+1}^{CK}\), then you can construct an SKIΩ tree corresponding to a rank- \(\omega_{\alpha+1}^{CK}\) Busy Beaver? How?

Same goes for " we can diagonalize through these ordinals". How do you diagonalize through the ordinals? What is at work here that doesn't work for \(\Phi(1,0)\)?