User blog comment:Ikosarakt1/Hardy hierarchy up to psi(K)/@comment-24509095-20140715144528/@comment-5150073-20140715162745

These rules, as I admitted in chat, work improperly past $$\chi(n,0)$$ for finite n.

There would be functions in the form $$H_{\psi_{\chi(\alpha,0)}(\beta)}(n)$$ for limit $$\alpha$$.

They work in stronger way: for example, $$\text{sup}\{\chi(1,0),\chi(2,0),\chi(3,0),\cdots\}$$ is $$\psi_{\chi(\omega,0)}(0)$$ and not just $$\chi(\omega,0)$$. This saves some structures.

"M" doesn't even follow simple definition of diagonalizers, because $$\text{sup}\{\chi(1,0),\chi(\chi(1,0),0),\chi(\chi(\chi(1,0),0),0),\cdots\}$$ limits to $$\psi_{\chi(M,0)}(0)$$ and not to $$\chi(M,0)$$.

Also, "K" is much more powerful - it's not just a diagonalizer of Mahlo-hierarchy of $$M_\alpha$$.