User blog comment:Deedlit11/Ordinal Notations V: Up to a weakly Mahlo cardinal/@comment-1605058-20130716063424/@comment-5529393-20130717212237

Yup. Note that the crucial property of these larger cardinals is that they are suffieciently large and far apart that we can't reach them from below using weaker methods, so far example we can't reach Omega using +, phi, or psi_Omega on countable ordinals, and we can't reach I using +, phi, a -> Omega_a, or psi_a with a <= I using ordinals less than I. Any ordinals that satisfy this property will work as well.