Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.

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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 $\omega_1^{ck}$, (Madore, 2017) 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. (Madore, 2017)

An ordinal $\alpha$ is *computably inaccessible*, also known as
*recursively inaccessible*, if it is admissible and a limit of
admissible ordinals. (Madore, 2017)

An ordinal $α$ is *recursively Mahlo* iff for any $α$-recursive function
$f : α → α$ there is an admissible $β < α$ closed under
$f$. (Madore, 2017)

There are also *recursively weakly compact* i.e. *$Π_3$-reflecting* or
*2-admissible* ordinals. (Madore, 2017)

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). (Madore, 2017)

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- Madore, D. (2017).
*A zoo of ordinals*. http://www.madore.org/ david/math/ordinal-zoo.pdf