User blog comment:Deedlit11/Extending the fast-growing hierarchy to nonrecursive ordinals/@comment-25418284-20130325203802/@comment-1605058-20130329145007

Let me call ordinal Xi-recursive if it's recursive in SKIΩ calculus. We proceed by induction. Every recursive ordinal is Xi-recursive. If  \(\alpha\)  isn't admissible, it's expressible as sum of admissible and recursive ordinals. Sum is Xi-recursive, so  \(\alpha\) is so too.

If \(\omega_\alpha^{CK}\) is Xi-recursive, then all ordinals below  \(\omega_{\alpha+1}^{CK}\) are Xi-recursive. By allowing  oracle operator to work on all these ordinals,  \(\omega_{\alpha+1}^{CK}\) is Xi-recursive.

If \( \alpha\) is limit and Xi-recursive, and all  \(\omega_{<\alpha}^{CK}\) are Xi-recursive, we can diagonalize through these ordinals, showing  \(\omega_\alpha^{CK}\) is Xi-recursive. This works all the way to \(\Phi (1,0)\)