User blog comment:Scorcher007/Large countable ordinal notation up to Z2 and ZFC/@comment-35470197-20181121071653/@comment-31580368-20181124063650

I think I understood the problem.

ω-th admissible ordinal also meaningless, but we can say about S[σ](ω) - limit of admissible. Then S[σ](ω+1) must be 1st admissible after limit of admissible. S[σ](ω×2) must be 2nd limit of admissible. S[σ](ω2) must be 2nd limit of admissible. S[σ](ωω) must be ω-th limit of admissible. Then S[σ](ε0) - ε0-th limit of admissible and it turns out that this is what I meant. I will correct the definitions later.