In set theory, an ordinal number α is an admissible ordinal if Lα is an admissible set (that is, a transitive model of Kripke–Platek set theory); in other words, α is admissible when α is a limit ordinal and Lα⊧Σ0-collection.[1][2]

The first two admissible ordinals are ω and ${\displaystyle \omega _{1}^{\mathrm {CK} }}$ (the least non-recursive ordinal, also called the Church–Kleene ordinal).[2] Any regular uncountable cardinal is an admissible ordinal.

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.[1] One sometimes writes ${\displaystyle \omega _{\alpha }^{\mathrm {CK} }}$ for the ${\displaystyle \alpha }$-th ordinal which is either admissible or a limit of admissibles; an ordinal which is both is called recursively inaccessible.[3] There exists a theory of large ordinals in this manner that is highly parallel to that of (small) large cardinals (one can define recursively Mahlo ordinals, for example).[4] But all these ordinals are still countable. Therefore, 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(Lα) mapping from γ onto α. If M is a standard model of KP, then the set of ordinals in M is an admissible ordinal.