From Cantor's Attic
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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 $\omega_1^{ck}$, 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 $L_{\omega_1^x}[x]\models\text{KP}$.

Computably inaccessible ordinal

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

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