User blog comment:Nayuta Ito/A suggestion to the extension of Psi's Letter Notation/@comment-30754445-20180202141551

In my system ωω[2] = ω2 as usual. You seem to have forgotten that in my notation, n is replaced by 10 when dealing with a limit ordinal.

So:

P2 = [ωω]2 = [ω2]10

This happens to be equal to [ω2+1]2 because:

[ω2+1]2 = [ω2]10 (for sucessor ordinals my notation recurses n-1 times rather than n times to maintain continuity)