Admissible

• Computably inaccessible ordinal
• Recursively Mahlo and further

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)

Computably inaccessible ordinal

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

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 $f$. (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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References

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

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