User blog comment:Scorcher007/Large countable ordinal notation up to Z2 and ZFC/@comment-11227630-20181121071343/@comment-11227630-20181121120328

1. The structure is still that rich, because your S[S[σ2+1](σ+1)] is actually S[S[σ2+1](σ+1)] = S[S[σ2+1](σ+1)+1], your S[S[σ2+1](σ+1)+1] is actually S[S[σ2+1](σ+1)+2], but your S[S[σ2+1](σ+1)+ω] is still actually S[S[σ2+1](σ+1)+ω].

5. "The final target of a stable chain is recursively inaccessible on the chained ordinals" means this ordinal is admissible and also limit of the chained ordinals, i.e. nonprojectable and admissible. So this concept is quite weak. S[S[S2[σσ'1]]] should still be П2-(St)-reflecting (П2-reflecting on the chained ordinals), S[S[S2[σσ'2]]] is П2-reflecting on П2-reflectings on the chained ordinals, S[S[S2[σσ'3]]] is П3-reflecting on the chained ordinals.