User blog comment:Deedlit11/Ordinal Notations V: Up to a weakly Mahlo cardinal/@comment-5029411-20130811125215/@comment-5529393-20140623140450

@Ikosarakt1: It actually works properly if you define the appropriate rules. For example, let's take the Bachmann-Howard notation to keep things simple. The way to define the fundamental sequence of psi(alpha) is:

psi(0)[0] = 0, of course.

cof (psi(alpha+1)) = w, and psi(alpha+1)[0] = psi(alpha)+1, psi(alpha+1)[n+1] = w^(psi(alpha+1)[n])

if cof(alpha) = w, then cof (psi(alpha)) = w, and psi(alpha)[n] = psi(alpha[n])

if cof(alpha) = Omega, then cof (psi(alpha)) = w, and psi(alpha)[0] = 0, psi(alpha)[n+1] = psi(alpha[psi(alpha)[n]])

If we take alpha = Omega, for instance, we take Omega[a] = a, and so

psi(Omega)[0] = 0, psi(Omega)[n+1] = psi(Omega[psi(Omega)[n]]) = psi(psi(Omega)[n])

so psi(Omega)[1] = psi (psi(Omega)[0]) = psi(0), psi(Omega)[2] = psi(psi(Omega[1])) = psi(psi(0), etc.

If we take alpha = Omega * 2, then Omega*2[a] = Omega + a, so

psi(Omega*2)[n+1] = psi(Omega*2[psi(Omega*2)[n]]) = psi(Omega + psi(Omega*2)[n])

so psi(Omega*2)[1] = psi(psi(Omega*2)[0]) = psi(Omega), psi(Omega*2)[2] = psi(psi(Omega*2)[1]) = psi(Omega + psi(Omega)), etc.

As you can see, the different values of the fundamental sequences for ordinals of uncountable cofinality translate perfectly into different rules for the fundamental sequences for psi(alpha). It's actually quite nice. psi_I(2)(alpha) works the same way, so:

if alpha has cofinality I(2), then psi_I(2)(alpha)[0] = 0, and psi_I(2)(alpha)[n+1] = psi_I(2)(alpha[psi_I(2)(alpha)[n]]).

So, since I(2)(a) = a, then psi_I(2)(I(2))[n+1] = psi_I(2)(psi_I(2)(I(2))[n]) and

psi_I(2)(I(2))[0] = 0, psi_I(2)(I(2))[1] = psi_I(2)(0), psi_I(2)(I(2))[2] = psi_I(2)(psi_I(2)(0)), etc.

which is exactly what you want.