Difference between revisions of "Reflecting ordinal"

Revision as of 08:07, 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 admissible ordinals $>\omega$. (theorem 1.8)[2]

TODO (equivalences, recursively Mahlo) (theorem 1.9)[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 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]