Admissible: Difference between revisions

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.<ref>ProbablyAdmissible inSets and Structures, Barwise, J., ''Perspectives in Logic'', Cambridge University somewherePress.</ref> The least admissible ordinal is [[Church-Kleene ordinal|<nowiki>\(\omega_1^{\mathrm{CK}}\)</nowiki>]], although some authors omit the axiom of infinity from KP and consider [[Omega|\(\omega\)]] to be admissible.