User blog comment:Scorcher007/Large countable ordinal notation up to Z2 and ZFC/@comment-11227630-20181121071343/@comment-31580368-20181121111612

1) If S[S[σ2+1](σ+1)+1] = S[S[σ2+1](σ+1)] than S[S[S[σ3+1](σ2+1)](σ+1)+1] = S[S[S[σ3+1](σ2+1)](σ+1)] and the structure is not as rich as I imagined.

2) Why there are two S's in "S[S[σω]]"? That to distinguish S[S[σω](1)] and S[S[σω]](1), but in accordance with the first point I'm not sure now that this is necessary.

3) Accordance with the first point maybe it's true. Anyway S[σω+1] = S[σω+1]

4) I do not know how to answer these questions.

5) Perhaps the concept of "the ordinal x that is x-ply-stable" is wrong. Then the further structure is again not as rich as I imagined. Then maybe it should be so: S[S[σα]] = S[S2[σσ](1undefined0)] = S[S2[σσ](S[S2[σσ](S[S2[σσ](...)])])] - 1st fixed point; S[S2[σσ](1undefined1)] - 2nd fixed point; S[S2[σσ'1]] is final target of a stable chain is recursively inaccessible on the chained ordinals; S[S2[σσ'2]] is final target of a stable chain is recursively Mahlo on the chained ordinals. Anyway S[S2[σσ+1]] must be (+1)-2-stable