Admissible

A set \(M\) is admissible if \((M,\in)\) is a model of Kripke-Platek set theory. An ordinal \(\alpha\) is admissible if there exists an admissible set \(M\) such that \(M\cap\textrm{Ord}=\alpha\). This definition of admissibility is equivalent to \(L_\alpha\vDash\textrm{KP}\), which is itself equivalent to \(\alpha > \omega\), \(\alpha\) being a limit ordinal and \(L_\alpha\) being closed under preimages of \(\alpha\)-recursively enumerable functions. The least admissible ordinal is \(\omega_1^{\mathrm{CK}}\), although some authors omit the axiom of infinity from KP and consider \(\omega\) to be admissible.