User blog comment:Deedlit11/Ordinal notations II: Up to the Bachmann-Howard ordinal/@comment-5150073-20130628190402

I saw that Chris Bird made the following definition of psi function:

$$psi(\alpha+1) = psi(\alpha)\uparrow\uparrow w$$ (i.e. a limit of power tower of $$psi(\alpha)$$'s) and all rules for theta function remain the same.