Admissible ordinal

In set theory, an admissible set is a transitive set $A\,$ such that $\langle A,\in \rangle$ is a model of Kripke–Platek set theory.

The smallest example of an admissible set is the set of hereditarily finite sets.

An ordinal number α is an admissible ordinal if L_α is an admissible set.

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 a limit of admissibles; 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 (one can define recursively Mahlo cardinals, for example). But all these ordinals are still countable.

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

Notice that α is an admissible ordinal if and only if α is a limit ordinal and there does not exist a γ<α for which there is a Σ₁(L_α) mapping from γ onto α. If M is a standard model of KP, then the set of ordinals in M is an admissible ordinal.

See also