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. They were introduced by Richter and Aczel in their 1974 paper "Inductive definitions and reflecting properties of admissible ordinals". (Kranakis 1985, Definable partitions and reflection properties for regular cardinals)
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}
(compare)
Properties
$Π_2$-reflecting ordinals are precisely the admissible ordinals $>\omega$ (class $\mathrm{Ad}$). (theorem 1.8)[2]
$\alpha$ is a limit of $X$ ($\alpha = \sup (X \cap \alpha)$) $\iff$ $\alpha$ is $\Pi_0^0$-reflecting on $X$ $\iff$ $\alpha$ is $\Sigma_2^0$-reflecting on $X$. (theorem 1.9 i)[2]
An ordinal is $\Pi_2^0$-reflecting on $X$ if it is recursively Mahlo on $X$. (theorem 1.9 ii)[2]
An ordinal is $\Pi_n^0$-reflecting on $X$ iff it is $\Sigma_{n+1}^0$-reflecting on $X$. (theorem 1.9 iii)[2]
When $Q$ is $\Pi_m^n$ for $m>2$, $\Pi_m^n$ for $n>0$, $\Sigma_m^n$ for $m>3$ or $\Sigma_m^n$ for $n>0$, an ordinal is $Q$-reflecting on $X$ iff it is $Q$-reflecting on $X \cap \mathrm{Ad}$. (theorem 1.9 iv)[2]
$\alpha$ is $Q$-reflecting on $X$ iff $\alpha$ reflects every $Q$-sentence of $\mathcal{L}_p$ on $X$, where $\mathcal{L}_p$ is the sublanguage of $\mathcal{L}$ with relation symbols only for the primitive recursive relations on sets. TODO: complete (theorem 1.10)[2]
$Π_3$-reflecting ordinals are precisely the 2-admissible ordinals (theorem 1.16)[2] They can be called recursively weakly compact.[3] More generally, $Π_{n+2}$-reflection is analogous to strong $Π_n^1$-indescribability for all $n>0$. (after definition 1.12)[2]
$(+1)$-stable ordinals are exactly the $Π^1_0$-reflecting (i.e., $Π_n$-reflecting for every $n ∈ ω$[3]) ordinals (Theorem 1.18). $({}^+)$-stable ordinals are exactly the $Π^1_1$-reflecting ordinals (Theorem 1.19).[2]
The least $\Sigma^1_1$-reflecting ordinal coincides with the least bad ordinal[3]. In comparison with weakenings of stable ordinals, this ordinal is less than the least $\alpha$ that's $\alpha^+ +1$-stable. (J. P. Aguilera, The Order of Reflection, 2019 arXiv preprint)
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