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

Since I don't understand the formalism above, let me ask a few questions. Based on my understanding of the $$\chi$$ function, it's just the $$I$$ function but extended similar to the Bachmann-Howard hierarchy. I know that $$\psi_{\chi(\alpha+1)}(0)$$ collapses to $$\chi(\alpha)_{\chi(\alpha)_{\chi(\alpha)_{...}}}$$ for some limit ordinal $$\alpha$$. Now, before this large cardinal, we have things like $$\chi(\alpha)_{\chi(\alpha)}$$, $$\chi(\alpha)_{\omega}$$, and, more importantly, $$\chi(\alpha)_2$$. My question is, how does one get from $$\psi_{\chi(\alpha)}$$ to $$\psi_{\chi(\alpha)_2}$$ for any limit ordinal $$\alpha$$? Similarly, how does one get from $$\psi_{\chi(M)}$$ to $$\psi_{\chi(M)_2}$$? Also, what do you mean by "A cardinal is 1-weakly inaccessible if it weakly inaccessible and a limit of weakly inaccessibles."?