Admissible ordinal

An admissible ordinal is any ordinal α such that L_α is a standard set model of Kripke-Platek set theory.

The first two admissible ordinals are ω and $\omega _{1}^{\mathrm {CK} }$ (the least non-recursive ordinal, also called the Church-Kleene ordinal).

By a theorem of Sacks, the countable admissible ordinals are exactly those which are constructed in a manner similar to the Church-Kleene ordinal but for Turing machines with oracles. One sometimes writes $\omega _{\alpha }^{\mathrm {CK} }$ for the $\alpha$-th ordinal which is either admissible or limit of admissible; an ordinal which is both is called recursively inaccessible: there exists a theory of large ordinals in this manner which is highly parallel to that of (small) large cardinals (we can define recursively Mahlo cardinals, for example). But note that we are still talking about countable ordinals here!

So admissible ordinals seem to be the recursive analogue of regular cardinals.

See also

• Large countable ordinals
• Inaccessible cardinal
• Constructible universe