Difference between revisions of "Reflecting ordinal"
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''Reflecting ordinals'' are large countable ordinals that show up in topics related to [[admissible|admissibility]] and [[reflecting cardinals|reflection principles]]. | ''Reflecting ordinals'' are large countable ordinals that show up in topics related to [[admissible|admissibility]] and [[reflecting cardinals|reflection principles]]. | ||
==Definition== | ==Definition== | ||
− | Let $\Pi$ denote its part of the Levy hierarchy. An ordinal $\alpha$ is $\Pi_n$''-reflecting'' | + | 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). <cite>Arai2019:FirstOrderReflection</cite><sup>page 1</sup><cite>RichterAczel1974:InductiveDefinitions</cite> |
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− | + | ([[Indescribable#Indescribable_on_a_set|compare]]) | |
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{{References}} | {{References}} |
Revision as of 01:39, 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]
(compare)
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