User blog comment:Deedlit11/Ordinal Notations IV: Up to a weakly inaccessible cardinal/@comment-5029411-20130722002210/@comment-5529393-20130722004925

$$\psi_I(0)$$ is the first fixed point of $$\alpha \mapsto \Omega_\alpha$$, so we can take

$$\psi_I(0) [0] = 0 $$

$$\psi_I(0) [n+1] = \Omega_{\psi_I(0) [n]}$$