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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 $$f$$[1]. A recursively Mahlo ordinal fixed in the context is sometimes denoted by $$\mu_0$$[1]. In particular, when one choose the least one, the least recursively Mahlo ordinal is denoted by $$\mu_0$$.[2] 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.[1]

## Properties

A set $$x$$ is recursively Mahlo if it is admissible, and for every $$\Sigma_0$$ formula $$\phi(y,z,\vec{p})$$, $$\forall\vec{p}\in x(\forall y\in x\exists z\in x\phi(y,z,\vec{p})\rightarrow\exists w\in x(w\text{ is admissible}\land\vec{p}\in w\land\forall y\in w\exists z\in w\phi(y,z,\vec{p})))$$.[3] 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.[4]

An ordinal $$\alpha$$ is recursively Mahlo if and only if it's $$\Pi_2$$-reflecting on the class of admissible ordinals[5]. As a result, recursively Mahlo ordinals are recursively inaccessible, recursively hyper-inaccessible, recursively hyper-hyper-inaccessible, etc.[6] (i.e., recursively $$\textrm{hyper}^n\!"\textrm -$$inaccessible with $$n\in\mathbb N$$).

## Sources

1. 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)
2. M. Rathjen, Ordinal notations based on a weakly Mahlo cardinal, Archive for Mathematical Logic, Volume 29, Issue 4, pp. 249--263, 1990.
3. Anton Setzer, "Universes in Type Theory Part I – Inaccessibles and Mahlo" (p.26)
4. M. Rathjen, Proof-theoretic analysis of KPM, Arch. Math. Logic 30 (1991) 377–403.
5. T. 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.
6. W. Richter, P. Aczel, Inductive Definitions and Reflecting Properties of Admissible Ordinals (1973) (p.13)
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