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Hello , I am wondering if the next admissible ordinal beyond $\omega_{1}^{ck}$ is still countable. Does someone know a paper or a book that explain what look like an admissible ordinal beyond $\omega_{1}^{ck}$.

Thanks !

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The Wikipedia page on admissible ordinals says that $\omega_1^{CK}(x)$ where $x$ is a real number is countable, and all countable admissible ordinals are those (and the limit of such). – Asaf Karagila Jun 9 '12 at 21:25
For $\alpha$ to be admissible means $L_\alpha$ satisfies a first order theory that any cardinal $\alpha$ (larger than $\omega$) is easily seen to satisfy. The set of $\alpha$ such that $L_\alpha\prec L_{\omega_1^V}$ is easily seen to be a club in the $\omega_1$ of $V$. In particular, the countable admissible ordinals are unbounded. – Andrés E. Caicedo Jun 10 '12 at 22:29
up vote 7 down vote accepted

The admissible ordinals are unbounded in $\omega_1$, so there are uncountably many countable admissible ordinals. Barwise's book, "Admissible Sets and Structures" is the standard reference on all things admissible. Beyond that, the proof theoretic literature contains some investigation of extensions of Kripke-Platek by "large cardinal" assumptions, beginning with KP+"there exists an admissible ordinal", and these theories are satisfied precisely by various larger admissible ordinals (and in all cases, there are countable ordinals which satisfy them). Various papers by Jäger, Pohlers, and Rathjen contain these theories.

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