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'' i&#x200b;f f&#x200b;or any formula φ(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>@paper{Arai2019:FirstOrderReflection,
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Let $\Pi$ denote its part of the Levy hierarchy. An ordinal $\alpha$ is $\Pi_n$''-reflecting'' if for any formula $\phi&#40;a)$ &#40;in a language such as "$\mathcal L_\in$ with parameters") that is $\Pi_n$, for all $a\in L_\alpha$, $L_\alpha\vDash\phi&#40;a)\rightarrow\exists&#40;\beta\in\alpha)&#40;L_\beta\vDash\phi&#40;a))$. &#40;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>
TITLE = {A simplified ordinal analysis of first-order reflection}
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AUTHOR = {Toshiyasu Arai}
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([[Indescribable#Indescribable_on_a_set|compare]])
PAGE = {1}
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URL = {https://arxiv.org/abs/1907.07611v1}
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}</cite>
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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

  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
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