An ordinal is recursively inaccessible if it is admissible and is a limit of admissible ordinals. Analogously, a set \(x\) is recursively inaccessible if it is admissible and \(\forall y\in x\exists z\in x(y\in z\land\,z\text{ is admissible})\).

Higher recursive inaccessibility can be defined similarly: an ordinal is recursively \(\alpha\)-inaccessible if it is admissible and is a limit of recursively \(\beta\)-inaccessible ordinals for all \(\beta<\alpha\). An ordinal \(\alpha\) is recursively hyperinaccessible if it is \(\alpha\)-inaccessible.

Properties

The following are equivalent for any ordinal \(\alpha\):

  1. \(\alpha\) is recursively inaccessible.
  2. \(L_{\alpha} \models \mathsf{KPi}\).
  3. \((\mathbb{N},\mathcal{P}(\mathbb{N}) \cap L_{\alpha},+,\times,<) \models \Delta^1_2\text{-}\mathsf{CA}+\mathsf{Bi}\).[1]
  4. \(L_{\alpha} \models \mathsf{KP}\beta\), where \(\mathsf{KP}\beta\) denotes \(\mathsf{KP}\omega\) augumented by Mostowski collapse lemma.

See also

Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · ordinal notation · transfinite induction
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)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\zeta,\Sigma,\gamma\) (ordinals on infinite time Turing machine) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Bachmann's function · Madore's function · Feferman's theta function · Buchholz's function · Extended Buchholz's function · Jäger's function · Rathjen's psi function · Rathjen's Psi function · Stegert's Psi function
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal
Classes: \(V\) · \(L\) · \(\textrm{On}\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)

Sources

  1. S.G. Simpson, Subsystems of Second-Order Arithmetic (second edition), Perspectives in Logic, ASL (2009), ISBN 978-0-521-88439-6
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