Reflecting ordinal
- Not to be confused with reflecting cardinals.
Reflecting ordinals are large countable ordinals that show up in topics related to admissibility and reflection principles.
Definition
Let $\Pi$ denote its part of the Levy hierarchy. An ordinal $\alpha$ is $\Pi_n$-reflecting if for any formula $\phi(a)$ (in a language such as "$\mathcal L_\in$ with parameters") that is $\Pi_n$, for all $a\in L_\alpha$, $L_\alpha\vDash\phi(a)\rightarrow\exists(\beta\in\alpha)(L_\beta\vDash\phi(a))$. (Note that by a formula such as $``\phi(a)"=\ulcorner a=a\urcorner$, the condition $a\in L_\beta$ becomes superfluous). [1]^{page 1}[2]^{definition 1.7}
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Properties
$Π_3$-reflecting ordinals are precisely 2-admissible ordinals (theorem 1.16)[2] They can be called recursively weakly compact. More generally, $Π_{n+2}$-reflection is analogous to strong $Π_n^1$-indescribability for all $n>0$. (after definition 1.12)[2][3]
References
- Arai, Toshiyasu. A simplified ordinal analysis of first-order reflection. , 2019. www bibtex
- Richter, Wayne and Aczel, Peter. Inductive Definitions and Reflecting Properties of Admissible Ordinals. Generalized recursion theory : proceedings of the 1972 Oslo symposium, pp. 301-381, 1974. www bibtex
- Madore, David. A zoo of ordinals. , 2017. www bibtex