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 \(\textsf{KP}\),Christoph Duchhardt, Thinning Operators and \(\Pi_4\)-Reflection, Munster university doctral thesis, 2008. Accessed 2021-04-16. 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.D. Madore, A Zoo of Ordinals #2.3 (2017, accessed 2020-11-18)
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. However, there are many ways of defining fundamental sequences (which are necessary to define the FGH), and busy beaver functions with oracles. In fact, a system of fundamental sequences such that the claim is false is known,T. Kihara, omega-1-ck.pdf. while a system of fundamental sequences such as the claim is true is not known.
==Properties==
*If \(\alpha\) is admissible, then there is no \(\Sigma_1\) projection with parameters in \(L_\alpha\) that maps some \(\gamma<\alpha\) to \(\alpha\)R. Jensen, The fine structure of the constructible hierarchy (1972, accessed 2020-11-19) (p.254)
*If \(\alpha\) is admissible, then \(L_\alpha\) satisfies a certain \(\Delta_1\)-separation schemaK. Devlin, An introduction to the fine structure of the constructible hierarchy (1974) (p.19).
==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 schemaE. Kranakis, Reflection and partition properties of admissible ordinals (p.216). Accessed 2021-03-27. 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 \(\Sigma_2\)-separationM. Rathjen, The Higher Infinite in Proof Theory (p.19). Accessed 2021-03-27.
== Sources ==
== See also ==
ja:許容順序数
Category:Transfinite ordinals
Category:Set theory