User blog comment:Scorcher007/Large countable ordinal notation up to Z2 and ZFC/@comment-35470197-20181121071653/@comment-35470197-20181124053331

First, you defined \(S[\sigma](\alpha)\) as the \(\alpha\)-th ordinal in the closure of the set of admissible countable ordinals. Therefore \(S[\sigma](\varepsilon_0)\) is the \(\varepsilon_0\)-th ordinal in the closure of the set of admissible countable ordinals.

In addition, if you assume the wrong property that \(S[\sigma](\varepsilon_0)\) is the \(\varepsilon_0\)-th admissible ordinal as you wrote, then it is simultaneously an admissible ordinal and a limit of admissible ordinals by the definition of the closure and the elementary fact that \(\varepsilon_0\) is a limit ordinal.

Therefore your assumption implies that \(S[\sigma](\varepsilon_0)\) is a recursively inaccessible ordinal. As a result, your assumption seems to be incorrect.

Similar for "1st fixed point of admissible".