Difference between revisions of "Reflecting ordinal"

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

$Π_3$-reflecting ordinals are precisely 2-admissible ordinals (theorem 1.16)[2] They can be called recursively weakly compact. More generally, $Π_{n+2}$-reflection is analogous to strong $Π_n^1$-indescribability for all $n>0$. (after definition 1.12)[2][3]

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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