Admissible ordinal

An ordinal is called admissible if \(L_\alpha\) is an admissible set, where \((L_\xi)_{\xi\in\textrm{Ord}}\) is the constructible hierarchy.

Here, a set \(M\) is said to be admissible (as a structure of set theory) if \((M,\in)\) is a model of \(\textsf{KP}\). Readers should be careful that some authors use \(\textsf{KP}\omega\) instead of \(\textsf{KP}\), and the admissibility for \(\textsf{KP}\omega\) is not equivalent to that for \(\textsf{KP}\). For example, the first admissible ordinal with respect to the admissibility for \(\textsf{KP}\) is \(\omega\), while the first admissible ordinal with respect to the admissibility for \(\textsf{KP} \omega\) is \(\omega_1^{\textrm{CK}}\) (also the second admissible ordinal with respect to the admissibility for \(\textsf{KP}\)). In this article, we deal with the convention of the admissibility for \(\textsf{KP}\).

The enumeration function of the class of admissible ordinals and limits of admissible ordinals is denoted by \(\alpha \mapsto \omega_{\alpha}^{\textrm{CK}}\). For a successor ordinal \(\alpha\), \(\omega_\alpha^\text{CK}\) is the \(1+\alpha\)-th admissible ordinal. However, \(\omega_\omega^\text{CK}\) and other ordinals of the form \(\omega_\alpha^\text{CK}\) with limit \(\alpha\) are not necessarily admissible although they are limits of admissibles. In the case that \(\omega_\alpha^\textrm{CK}\) is both admissible and a limit of admissibles, it's also called a recursively inaccessible ordinal.

It was believed in this community without proofs that \(f_{\omega_\alpha^\text{CK}}(n)\) is approximated to \(\Sigma_\alpha(n)\), the \(\alpha\)-th order busy beaver function using some unspecified system of fundamental sequences. Due to many ways of defining fundamental sequences and busy beaver functions with oracles it may be not true. In fact, the existence of a system of fundamental sequences such that the property is false is known. Therefore the belief is wrong.

Properties

 * If \(\alpha\) is admissible, then there is no \(\Sigma_1\) projection with parameters in \(L_\alpha\) that maps some \(\gamma<\alpha\) to \(\alpha\)
 * If \(\alpha\) is admissible, then \(L_\alpha\) satisfies a certain \(\Delta_1\)-separation schema.

Extension
An extension of the concept of admissibility is \(\Sigma_n\)-admissibility for natural \(n\). An ordinal \(\alpha\) is \(\Sigma_n\)-admissible if \(L_\alpha\) is rudimentary closed and satisfies the \(\Sigma_n\)-collection schema. Note that ordinals which are \(\Sigma_n\)-admissible need not satisfy \(\Sigma_n\)-separation, for example the least \(\Sigma_2\)-admissible ordinal doesn't satisfy \(\Sigma_2\)-separation.