User blog comment:Kyodaisuu/Mashimo function/@comment-1605058-20140706065714/@comment-5150073-20140706210502

It's the first ordinal larger than any recursive extension of $$a \mapsto \omega_a^\text{CK}$$. We can define $$\psi_\text{CK}(\alpha)$$ function which indicates $$1+\alpha$$-th fixed point of $$a \mapsto \omega_\alpha^\text{CK}$$ and define diagonalizers inside it. So we go through things like:

$$\psi_\text{CK}(\Omega)$$

$$\psi_\text{CK}(I)$$

$$\psi_\text{CK}(M)$$

$$\psi_\text{CK}(K)$$

So we suppose that $$\omega_\alpha^\text{CK}[n]$$ is computable in a certain system, as long as $$\alpha$$ is defined itself. Then make recursion around it, using $$\psi_\text{CK}$$, $$\theta_\text{CK}$$ and similar functions. The limit of all this would be which I call $$\omega_{1,2}^\text{CK}$$.