# Admissible

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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## References

Main library