Large Countable Ordinal - Beyond Recursive Ordinals - Admissible Ordinals

Admissible Ordinals

The Church-Kleene ordinal is again related to Kripke-Platek set theory, but now in a different way: whereas the Bachmann-Howard ordinal (described above) was the smallest ordinal for which KP does not prove transfinite induction, the Church-Kleene ordinal is the smallest α such that the construction of the Gödel universe, L, up to stage α, yields a model of KP. Such ordinals are called admissible, thus is the smallest admissible ordinal (beyond ω in case the axiom of infinity is not included in KP).

By a theorem of Sacks, the countable admissible ordinals are exactly those constructed in a manner similar to the Church-Kleene ordinal but for Turing machines with oracles. One sometimes writes for the -th ordinal that is either admissible or a limit of admissible.