User blog comment:Deedlit11/Ordinal Notations V: Up to a weakly Mahlo cardinal/@comment-24509095-20140509071457/@comment-5529393-20140605091751

Ikosarakt1: I believe the natural definition for $$\chi_M_2 (0)$$ will be $$I_{M+1}$$. Then $$\chi_M_2 (0,\alpha)$$ will be $$I_{M+1+\alpha}$$, $$\chi_M_2(1,\alpha)$$ will be the 1-weakly inaccessible cardinals above $$I_{M+1}$$, and so on.

King2218: I'm not sure what you mean by "how in the world can we get from $$\chi(M)$$ to $$\chi(\chi(\ldots(\chi(\chi(M) + 1)\ldots))$$".   The values of $$\chi$$ are just various levels of diagonalizers.  Just like $$\Omega_\alpha$$ are diagonalizers. I diagonalizes over the $$\Omega_\alpha$$, $$I_2$$ diagonalizes over the $$\Omega_\alpha$$ above I, and so on. Then I(1,0) diagonalizes over the $$I_\alpha$$, and we continue in ths fashion for higher levels of inaccessibles. The $$\chi$$ function is just a fancier notation that extends the I notation, just as the Bachmann-Howard notation extends the Extended Veblen notation.  So we just accept these diagonalizers as given, than use them to collapse ordinals at various levels.

Please ask more questions if you are still confused.