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). <bibtex>@paper{Arai2019:FirstOrderReflection,
+
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,
  TITLE = {A simplified ordinal analysis of first-order reflection}
+
TITLE = {A simplified ordinal analysis of first-order reflection}
  AUTHOR = {Toshiyasu Arai}
+
AUTHOR = {Toshiyasu Arai}
  PAGE = {1}
+
PAGE = {1}
  URL = {https://arxiv.org/abs/1907.07611v1}
+
URL = {https://arxiv.org/abs/1907.07611v1}
}</bibtex>
+
}</cite>
  
 
{{References}}
 
{{References}}

Revision as of 11:59, 14 July 2021

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 i​f f​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). [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]

References

  1. Arai, Toshiyasu. A simplified ordinal analysis of first-order reflection. , 2019. www   bibtex
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