# Admissible ordinal explained

In set theory, an ordinal number *α* is an **admissible ordinal** if L_{α} 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 L_{α} ⊧ Σ_{0}-collection.^{[1]} ^{[2]}

The first two admissible ordinals are ω and

(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

for the

-th ordinal that is either admissible or a limit of admissibles; an ordinal that 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 Σ_{1}(L_{α}) mapping from *γ* onto *α*.^{[5]} If *M* is a standard model of KP, then the set of ordinals in *M* is an admissible ordinal.

## See also

## Notes and References

- . See in particular p. 265.
- .
- . See in particular p. 560.
- .
- K. Devlin, An introduction to the fine structure of the constructible hierarchy (1974) (p.38). Accessed 2021-05-06.