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An ordinal $\gamma$ is admissible if the $L_\gamma$ level of the constructible universe satisfies the Kripke-Platek axioms of set theory.

The smallest admissible ordinal is $\omega_1^{ck}$,[1] the least non-computable ordinal. More generally, for any real $x$, the least ordinal not computable from $x$ is denoted $\omega_1^x$, and is also admissible. Indeed, one has $L_{\omega_1^x}[x]\models\text{KP}$.

The smallest limit of admissible ordinals, $\omega_\omega^{ck}$, is not admissible.[1]

Computably inaccessible ordinal

An ordinal $\alpha$ is computably inaccessible, also known as recursively inaccessible, if it is admissible and a limit of admissible ordinals.[1]

Recursively Mahlo and further

An ordinal $α$ is recursively Mahlo iff for any $α$-recursive function $f : α → α$ there is an admissible $β < α$ closed under $f$.[1]

There are also recursively weakly compact i.e. $Π_3$-reflecting or 2-admissible ordinals.[1]

The smallest $Σ_2$-admissible ordinal is greater then the smallest nonprojectible ordinal and weaker variants of stable ordinals but smaller than the height of the minimal model of ZFC (if it exists).[1]

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  1. Madore, David. A zoo of ordinals. , 2017. www   bibtex
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