Reflecting ordinal

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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

$Π_2$-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]

References

  1. Arai, Toshiyasu. A simplified ordinal analysis of first-order reflection. , 2019. www   bibtex
  2. 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
  3. Madore, David. A zoo of ordinals. , 2017. www   bibtex
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