# Difference between revisions of "Admissible"

From Cantor's Attic

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The smallest admissible ordinal is [[Church-Kleene | $\omega_1^{ck}$]], 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 $L_{\omega_1^x}[x]\models\text{KP}$. | The smallest admissible ordinal is [[Church-Kleene | $\omega_1^{ck}$]], 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 $L_{\omega_1^x}[x]\models\text{KP}$. | ||

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+ | == Computably inaccessible ordinal == | ||

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+ | 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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## Revision as of 08:44, 30 December 2011

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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