Proof of existence of recursively inaccessible and Mahlo ordinals As in title - I'm looking for a proof of the existence of a countable recursively inaccessible or recursively Mahlo ordinals, especially the first one. When looking for it in all the papers I stumbled across their existence wasn't questioned, and no proofs were referenced. On Wikipedia (where I found the first mention of provability of their existence in ZFC) there is also no direct reference.
Thank you in advance for any help!
 A: An ordinal $\xi$ is recursively inaccessible when it is an admissible limit of admissible ordinals. To be admissible means that $L_\alpha$ is a model of the Kripke-Platek axioms of set theory. It is not difficult to see that $\omega_1$ itself is admissible, since $L_{\omega_1}\models\text{KP}$, and by taking elementary substructures, it follows that there is a closed unbounded set of ordinals $\alpha$ with $L_\alpha\prec L_{\omega_1}$. Thus, $\omega_1$ is recursively inaccessible, and indeed a limit of recursively inaccessible ordinals, because this is reflected to $\alpha$ whenever $L_\alpha\prec L_{\omega_1}$. So we have an abundance of these ordinals.
I'm not exactly sure what you mean by recursively Mahlo, but I suppose you want that the set of smaller admissible ordinals $\alpha$ is stationary with respect to definable clubs of some complexity. But this property is also true of true $\omega_1$, since as we said, there is a full closed unbounded set of countable admissible ordinals $\alpha$. And so it follows again by taking elementary substructures $L_\alpha\prec L_{\omega_1}$ that there will be many countable ordinals with this property.
