Recursively Mahlo ordinal

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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 \(f\). 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 \(n\in\mathbb N\)).

See also

• Kripke–Platek set theory

Ordinals, ordinal analysis, and set theory

Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation · Absolute infinity
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Model Theoretic Concepts: structure · elementary embedding
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\eta_0\) · \(\Gamma_0\) (Feferman–Schütte ordinal) · \(\varphi(1,0,0,0)\) (Ackermann ordinal) · \(\psi_0(\Omega^{\Omega^\omega})\) (small Veblen ordinal) · \(\psi_0(\Omega^{\Omega^\Omega})\) (large Veblen ordinal) · \(\psi_0(\varepsilon_{\Omega + 1}) = \psi_0(\Omega_2)\) (Bachmann-Howard ordinal) · \(\psi_0(\Omega_\omega)\) with respect to Buchholz's ψ · \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (Takeuti-Feferman-Buchholz ordinal) · \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) (countable limit of Extended Buchholz's function)‎ · \(\omega_1^\mathfrak{Ch}\) (Omega one of chess) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\gamma,\zeta,\Sigma\) (Infinite time Turing machine ordinals) · gap ordinal · List of countable ordinals
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