User blog comment:Deedlit11/Ordinal Notations V: Up to a weakly Mahlo cardinal/@comment-5029411-20130811125215/@comment-5150073-20140623152613

That would be okay, expect cof(alpha) function. Formally, it's an uncomputable function itself (no Turing machine is able to determine cofinality of the ordinal), and thus $$\psi$$, defined in this way, becomes uncomputable.