An ordinal $\gamma$ is *admissible* if the $L_\gamma$ level of the [[Constructible universe]] satisfies the [Kripke-Platek](Kripke-Platek.md "Kripke-Platek") axioms of set theory. The smallest admissible ordinal is the [[Church-Kleene]] ordinal $\omega_1^{ck}$, {% cite Madore2017 %} 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. {% cite Madore2017 %} ## Computably inaccessible ordinal An ordinal $\alpha$ is *computably inaccessible*, also known as *recursively inaccessible*, if it is admissible and a limit of admissible ordinals. {% cite Madore2017 %} ## 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 $f$. {% cite Madore2017 %} There are also *recursively weakly compact* i.e. *$Π_3$-reflecting* or *2-admissible* ordinals. {% cite Madore2017 %} The smallest $Σ_2$-admissible ordinal is greater then the smallest <a href="Nonprojectible" class="mw-redirect" title="Nonprojectible">nonprojectible</a> ordinal and weaker variants of [stable](Stable.md "Stable") ordinals but smaller than the height of the <a href="Transitive_ZFC_model" class="mw-redirect" title="Transitive ZFC model">minimal model of ZFC</a> (if it exists). {% cite Madore2017 %} This article is a stub. Please help us to improve Cantor's Attic by adding information.