User blog comment:Deedlit11/Ordinal Notations V: Up to a weakly Mahlo cardinal/@comment-24509095-20140509071457/@comment-5529393-20140605101016

$$f_{\psi_{\chi(\alpha)}(\chi(\alpha))}(n)$$ is meaningless; $$\psi_{\chi(\alpha)}(\chi(\alpha))$$ is not an appropriate index for the fast-growing hierarchy because it is an uncountable cardinal. An index for f must be a countable ordinal, so it should be of the form $$\psi_{\Omega_1}(\alpha)$$ for some ordinal $$\alpha$$ (possibly uncountable).

$$\chi(\alpha)$$ does not have a countable fundamental sequence, as it is an inaccessible regular cardinal, so the shortest sequence whose limit is $$\chi(\alpha)$$ is of length $$\chi(\alpha)$$. $$\chi(\alpha)$$ is meant to be used simply for collapsing to smaller ordinals, not for taking the fundamental sequence.

The reason for having $$\chi_{M_2}(0,0)$$ be $$I_{M+1}$$ rather than $$\Omega_{M+1}$$ is because $$\Omega_\alpha$$ is already in the list of functions used in our notation, so we can just represent $$\Omega_{M+1}$$ directly rather than collapsing some larger ordinal.