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 \(f\)M. Rathjen, Collapsing functions based on recursively large ordinals: A well–ordering proof for KPM (p.3) Archive for Mathematical Logic 33, 35-55 (1994). 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 \(\mu_0\).M. Rathjen, Ordinal notations based on a weakly Mahlo cardinal, Archive for Mathematical Logic, Volume 29, Issue 4, pp. 249--263, 1990. 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 \(\phi(y,z,\vec{p})\), \(\forall\vec{p}\in x(\forall y\in x\exists z\in x\phi(y,z,\vec{p})\rightarrow\exists w\in x(w\text{ is admissible}\land\vec{p}\in w\land\forall y\in w\exists z\in w\phi(y,z,\vec{p})))\).Anton Setzer, "Universes in Type Theory Part I – Inaccessibles and Mahlo" (p.26) For any ordinal \(\alpha\), the following are equivalent:
# \(\alpha\) is a recursively Mahlo ordinal.
# \(L_\alpha\) is a recursively Mahlo set.
# \(L_\alpha \models \textsf{KPM}\), i.e. \(L_{\alpha}\) is a union of admissible sets and is a recursively Mahlo set.M. Rathjen, Proof-theoretic analysis of KPM, Arch. Math. Logic 30 (1991) 377–403.
An ordinal \(\alpha\) is recursively Mahlo if and only if it's \(\Pi_2\)-reflecting on the class of admissible ordinalsT. Arai, Proof theory for theories of ordinals—I: recursively Mahlo ordinals, Annals of Pure and Applied Logic, Volume 122, Issues 1–-3, Pages 1--85, 2003.. As a result, recursively Mahlo ordinals are recursively inaccessible, recursively hyper-inaccessible, recursively hyper-hyper-inaccessible, etc.W. Richter, P. Aczel, Inductive Definitions and Reflecting Properties of Admissible Ordinals (1973) (p.13) (i.e., recursively \(``\textrm{hyper}^n\!"\textrm -\)inaccessible with \(n\in\mathbb N\)).
== See also ==
* Kripke–Platek set theory
== Sources ==
ja:再帰的マーロ順序数
Category:Transfinite ordinals
Category:Set theory