User blog comment:Kyodaisuu/Mashimo function/@comment-1605058-20140706065714/@comment-5150073-20140707091444

Sorry, I can't imagine how $$\omega_2^\text{CK}$$ can be defined from $$\omega_1^\text{CK}$$ without considering recursive extensions and diagonalizers. The definition here is extremely confusing, so I can't use it.

As for Turing machines, they can handle ordinals only up to $$\omega_1^\text{CK}$$, so of course they can't handle $$\omega_\alpha^\text{CK}$$. Same goes to Kleene's O.

The formal definition of $$\omega_{1,2}^\text{CK}$$ is:


 * If $$\psi(D_n)$$ is recursive, then $$\psi_\text{CK}(D_n) < \omega_{1,2}^\text{CK}$$.
 * $$\omega_{1,2}^\text{CK}$$ is the first ordinal for which we can create such an inequality.