User blog comment:Scorcher007/Large countable ordinal notation up to Z2 and ZFC/@comment-35470197-20181121071653/@comment-35470197-20181121133540

Well, the notation of \(\omega_{\alpha}^{\textrm{CK}}\) has two distinct meanings in the same context. Please tell me whether \(S[\sigma](\alpha)\) the \(\alpha\)-th ordinal in the closure of the set of admissible countable ordinals or not. If the answer is no, please clarify the definition of \(\omega_{\alpha}^{\textrm{CK}}\). I am asking this question because your table seems to conflict itself by mixing the two distinct meanings.