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 !
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 ! 


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 KripkePlatek 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. 

