User blog comment:Deedlit11/Ordinal Notations V: Up to a weakly Mahlo cardinal/@comment-5150073-20130809111936/@comment-5150073-20130809202944

So, how these n-inaccessible cardinal work? I think that they could be described informally as follows:

We start with $$\Omega$$, then we define the functions $$psi_\alpha$$ and cardinals $$Omega_\alpha$$. The limit of defining the ordinals this way: $$\Omega_{\Omega_{\cdots}}$$ requires the new "superfunction": $$\psi_I(\alpha)$$. This "I" works like the new kind of diagonalizator, and we can start to define $$psi_\alpha$$, where $$\alpha$$ can contains $$I_\alpha$$. Then limit of this notation will requires third type of diagonalization: we can express it as I(3), and $$\Omega$$ and I as I(1) and I(2) respectively.