User blog comment:Scorcher007/Large countable ordinal notation up to Z2 and ZFC/@comment-35470197-20181121071653/@comment-35470197-20181124074149

Well, \(\omega\)-th admissible ordinal is not meaningless. It is just \(S[\sigma](\omega+1)\), as you correctly defined \(S[\sigma](\omega)\) as the fist limit of admissible ordinals. The problem is just you are applying to distinct conventions, i.e. the enumeration of admissible countable ordinals and the enumeration of the closure of the set of admissible countable ordinals.

Also, the reason why "1st fixed point of admissible" in your table is incorrect is because the precise "1st fixed point of admissible" is automatically admissible and a limit of admissible ordinals, because every admissible ordinal is a limit ordinal. So what you meant in your mind would be "1st fixed point of \(S[\sigma]\)".