Recursively Mahlo ordinal

An ordinal \(\alpha\) is recursively Mahlo if it is admissible and for every \(\alpha\)-recursive function \(f \colon \alpha\to\alpha\), there exists \(\beta<\alpha\) such that \(\beta\) is admissible and closed under \(f\). A recursively Mahlo ordinal fixed in the context is sometimes denoted by \(\mu_0\). In particular, when one choose the least one, the least recursively Mahlo ordinal is denoted by \(\mu_0\). However, it does not mean that \(\mu_0\) always denotes the least recursively Mahlo ordinal.

\(\alpha\)-recursive Functions
\(\alpha\)-recursive functions are defined for admissible ordinals \(\alpha\). A function \(f\) is \(\alpha\)-recursive if its graph is definable on \(L\)\(_\alpha\) using a \(\Delta_1\) formula.

Properties
A set \(x\) is recursively Mahlo if it is admissible, and for every \(\Sigma_0\) formula \(\phi(y,z,\vec{p})\), \(\forall\vec{p}\in x(\forall y\in x\exists z\in x\phi(y,z,\vec{p})\rightarrow\exists w\in x(w\text{ is admissible}\land\vec{p}\in w\land\forall y\in w\exists z\in w\phi(y,z,\vec{p})))\). For any ordinal \(\alpha\), the following are equivalent:
 * 1) \(\alpha\) is a recursively Mahlo ordinal.
 * 2) \(L_\alpha\) is a recursively Mahlo set.
 * 3) \(L_\alpha \models \textsf{KPM}\), i.e. \(L_{\alpha}\) is a union of admissible sets and is a recursively Mahlo set.

An ordinal \(\alpha\) is recursively Mahlo if and only if it's \(\Pi_2\)-reflecting on the class of admissible ordinals. As a result, recursively Mahlo ordinals are recursively inaccessible, recursively hyper-inaccessible, recursively hyper-hyper-inaccessible, etc. (i.e., recursively \(``\textrm{hyper}^n\!"\textrm -\)inaccessible with \(n\in\mathbb N\)).