User blog comment:Scorcher007/Large countable ordinal notation up to Z2 and ZFC/@comment-35470197-20181121071653/@comment-31580368-20181121162132

ωε 0 CK is not recursibly inaccessible. I think there is an analogy with the cardinals. S[σ](0) - ω, S[σ](0) - ω1, S[σ](ε0) - ωε 0, e t.c. up to α↦S[σ](α), but we think not about regular and singular, but about admissible and limit of admissible.