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

1. S[S[σ2+1](σ+1)] is identical to S[S[σ2+1](σ+1)+1] due to the transitivity of ≺1.

2. S[S[σω](1)](ω) is ω-th nonprojectable limit of nonprojectables; S[S[σω](2)] is nonprojectable limit of nonprojectable limits of nonprojectables; etc. then S[S[σω](ω)] is the level ω of them.

3. Beyond S[S[σα]], "the ordinal x that is x-ply-stable" is S[S[σσ]]. Then 4. П2-(St)-reflecting means the final target of a stable chain is recursively Mahlo on the chained ordinals. It should be stronger than what you wrote "inaccessible limit of ...-ply stable".
 * what is the ordinal x that is stable up to another ordinal y that is y-ply-stable?
 * what is the ordinal x that is stable up to another ordinal y that is stable up to another ordinal z that is z-ply-stable?
 * what is the ordinal x that is stable up to a nonprojectable y that is y-ply-stable?
 * what is the ordinal x that is x-ply (ordinal y that is y-ply-stable)-stable?
 * what is the x-ply x-stable ordinal? (the length of stable chain is equal to the final target of the stable chain)