User blog comment:Deedlit11/Ordinal Notations V: Up to a weakly Mahlo cardinal/@comment-5029411-20130811125215/@comment-5529393-20140623153950

No, the cofinality of the ordinals in a notation is certainly computable if the notation is. Let's take the notation up to the Bachmann-Howard ordinal again. The rules are:

a_1 + a_2 + ... + a_n has cofinality equal to the cofinality of a_n.

0 has cofinality 0.

w^0 has cofinality 1.

if a has cofinality 1, w^a has cofinality w.

if a has cofinality w, w^a has cofinality w.

if a has cofinality Omega, w^a has cofinality Omega.

psi(a) has cofinality w for all a.

So for this notation, it's pretty simple. For more complicated notations, it gets more difficult, as does defining fundamental sequences. But it's definitely easier to define fundamental sequences if you take advantage of cofinality.