Admissible ordinal

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 ⊧ Σ₀-collection.^[1][2] The term was coined by Richard Platek in 1966.^[3]

The first two admissible ordinals are ω and ${\displaystyle \omega _{1}^{\mathrm {CK} }}$ (the least nonrecursive 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 that is either admissible or a limit of admissibles; an ordinal that is both is called recursively inaccessible.^[4] 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).^[5] 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(L_α) mapping from γ onto α.^[6] ${\displaystyle \alpha }$ is an admissible ordinal iff there is a standard model ${\displaystyle M}$ of KP whose set of ordinals is ${\displaystyle \alpha }$, in fact this may be take as the definition of admissibility.^[7][8] The ${\displaystyle \alpha }$th admissible ordinal is sometimes denoted by ${\displaystyle \tau _{\alpha }}$^{[9][8]p. 174} or ${\displaystyle \tau _{\alpha }^{0}}$.^[10]

The Friedman-Jensen-Sacks theorem states that countable ${\displaystyle \alpha }$ is admissible iff there exists some ${\displaystyle A\subseteq \omega }$ such that ${\displaystyle \alpha }$ is the least ordinal not recursive in ${\displaystyle A}$.^[11] Equivalently, for any countable admissible ${\displaystyle \alpha }$, there is an ${\displaystyle A\subseteq \mathbb {N} }$ making ${\displaystyle \alpha }$ minimal such that ${\displaystyle \langle L_{\alpha },\in ,A\rangle }$ is an admissible structure.^{[12]p. 264}