User blog comment:Deedlit11/Ordinal Notations V: Up to a weakly Mahlo cardinal/@comment-10429372-20130721064019/@comment-5529393-20130722001217

psi_{I(1,0)} (a) is the ath cardinal b such that c < b implies I(c) < b.

psi_{I(1,0)} (I(1,0)) is the smallest cardinal b such that b = psi_{I(1,0)} (b).

For a < I(1,0), psi_a (I(1,0)) is the limit of psi_a(psi_{I(1,0)}(b)) is b goes to I(1,0).

So psi_a (I(1,0)) has cofinality w, and

psi_a (I(1,0)) [n] = psi_a (z_n), where z_0 = 0 and z_{n+1} = psi_{I(1,0)}(z_n).

For completeness sake, here are fundamental sequences for psi_{I(1,0)}(a):

If a is limit, cof (psi_{I(1,0)}(a)) = cof (a), and

psi_{I(1,0)}(a) [n] = psi_{I(1,0)} (a[n]).

psi_{I(1,0)}(0) has cofinality w and

psi_{I(1,0)}(0) [0] = 0

psi_{I(1,0)}(0) [n+1] = I(psi_{I(1,0)}(0) [n])

psi_{I(1,0)}(a+1) has cofinality w and

psi_{I(1,0)}(a+1) [0] = psi_{I(1,0)}(a) + 1

psi_{I(1,0)}(a+1) [n+1] = I(psi_{I(1,0)}(a+1) [n])