Difference between revisions of "Reflecting ordinal"
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==Properties== | ==Properties== | ||
− | $Π_2$-reflecting ordinals are precisely the [[admissible]] ordinals $>\omega$. (theorem 1.8)<cite>RichterAczel1974:InductiveDefinitions</cite> | + | $Π_2$-reflecting ordinals are precisely the [[admissible]] ordinals $>\omega$ (class $\mathrm{Ad}$). (theorem 1.8)<cite>RichterAczel1974:InductiveDefinitions</cite> |
− | + | $\alpha$ 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)<cite>RichterAczel1974:InductiveDefinitions</cite> | |
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+ | An ordinal is $\Pi_2^0$-reflecting on $X$ if it is recursively [[Mahlo]] on $X$. (theorem 1.9 ii)<cite>RichterAczel1974:InductiveDefinitions</cite> | ||
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+ | An ordinal is $\Pi_n^0$-reflecting on $X$ iff it is $\Sigma_{n+1}^0$-reflecting on $X$. (theorem 1.9 iii)<cite>RichterAczel1974:InductiveDefinitions</cite> | ||
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+ | 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)<cite>RichterAczel1974:InductiveDefinitions</cite> | ||
$\alpha$ 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 | $\alpha$ 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)<cite>RichterAczel1974:InductiveDefinitions</cite> | symbols only for the primitive recursive relations on sets. ''TODO: complete'' (theorem 1.10)<cite>RichterAczel1974:InductiveDefinitions</cite> | ||
− | $Π_3$-reflecting ordinals are precisely 2-[[admissible]] ordinals (theorem 1.16)<cite>RichterAczel1974:InductiveDefinitions</cite> They can be called ''recursively [[weakly compact]]''.<cite>Madore2017:OrdinalZoo</cite> More generally, $Π_{n+2}$-reflection is analogous to strong $Π_n^1$-[[indescribable|indescribability]] for all $n>0$. (after definition 1.12)<cite>RichterAczel1974:InductiveDefinitions</cite> | + | $Π_3$-reflecting ordinals are precisely the 2-[[admissible]] ordinals (theorem 1.16)<cite>RichterAczel1974:InductiveDefinitions</cite> They can be called ''recursively [[weakly compact]]''.<cite>Madore2017:OrdinalZoo</cite> More generally, $Π_{n+2}$-reflection is analogous to strong $Π_n^1$-[[indescribable|indescribability]] for all $n>0$. (after definition 1.12)<cite>RichterAczel1974:InductiveDefinitions</cite> |
$(+1)$-[[stable]] ordinals are exactly the $Π^1_0$-reflecting (i.e., $Π_n$-reflecting for every $n ∈ ω$<cite>Madore2017:OrdinalZoo</cite>) ordinals (Theorem 1.18). $({}^+)$-stable ordinals are exactly the $Π^1_1$-reflecting ordinals (Theorem 1.19).<cite>RichterAczel1974:InductiveDefinitions</cite><!--Page 18 in the PDF, with label 16--> | $(+1)$-[[stable]] ordinals are exactly the $Π^1_0$-reflecting (i.e., $Π_n$-reflecting for every $n ∈ ω$<cite>Madore2017:OrdinalZoo</cite>) ordinals (Theorem 1.18). $({}^+)$-stable ordinals are exactly the $Π^1_1$-reflecting ordinals (Theorem 1.19).<cite>RichterAczel1974:InductiveDefinitions</cite><!--Page 18 in the PDF, with label 16--> | ||
{{References}} | {{References}} |
Revision as of 22:32, 14 May 2022
- 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}
(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]
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