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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.