User blog comment:Kyodaisuu/Mashimo function/@comment-1605058-20140706065714/@comment-5529393-20140707185516

I get the intuitive motivation behind the term "recursive extension" but we have to be careful that we have an explicit definition.

For a given $$\omega_\alpha^{\text{CK}}$$, we can talk about ordinals can be represented using Turing machines with an oracle for $$\omega_\alpha^{\text{CK}}$$. We are relying on the Wikipedia article to tell us that this will be $$\omega_{\alpha+1}^{\text{CK}}$$. But how do we define a "recursive extension" of $$\alpha \mapsto \omega_\alpha^\text{CK}$$? If we simply add a notation for that function (say $$O(7^n) = \omega_{O(n)}^\text{CK}$$), we will just top out at the first fixed point of $$\alpha \mapsto \omega_\alpha^\text{CK}$$. I don't think this is what Ikosarakt1 had in mind, so we need to somehow include fixed points as well. But what is a good formal definition of "all fixed point extensions of this function"? If we can solve this, then we can have a good definition of $$\omega_{1,2}^\text{CK}$$.