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!



Proof of existence of recursively inaccessible and Mahlo ordinals
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.


