Recursively inaccessible ordinal

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 \(\alpha\):
# \(\alpha\) is recursively inaccessible.
# \(L_{\alpha} \models \mathsf{KPi}\).
# \((\mathbb{N},\mathcal{P}(\mathbb{N}) \cap L_{\alpha},+,\times,<) \models \Delta^1_2\text{-}\mathsf{CA}+\mathsf{Bi}\).S.G. Simpson, Subsystems of Second-Order Arithmetic (second
edition), Perspectives in Logic, ASL (2009), ISBN 978-0-521-88439-6
# \(L_{\alpha} \models \mathsf{KP}\beta\), where \(\mathsf{KP}\beta\) denotes \(\mathsf{KP}\omega\) augumented by Mostowski collapse lemma.
== See also ==
* Kripke–Platek set theory‎‎
== Sources ==
ja:再帰的到達不能順序数
Category:Transfinite ordinals
Category:Set theory