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diff --git a/ordinal-zoo.tex b/ordinal-zoo.tex index 54b4f82..86cde74 100644 --- a/ordinal-zoo.tex +++ b/ordinal-zoo.tex @@ -189,14 +189,20 @@ subsets of $\omega$, and they are also exactly the subsets recursive \ordinal $\omega_\omega^{\mathrm{CK}}$: the smallest limit of admissibles. This ordinal is not admissible. This is the smallest $\alpha$ such that $L_\alpha \cap \mathscr{P}(\omega)$ is a model of -$\Pi^1_1$-comprehension. +$\Pi^1_1$-comprehension (cf. \cite[theorem VII.1.8 on p. 246 and + theorem VII.5.17 on p. 292 and notes to §VII.5 on + p. 293]{Simpson2009}). \ordinal\label{RecursivelyInaccessible} The smallest recursively inaccessible ordinal: this is the smallest ordinal which is admissible and limit of admissibles. This is the smallest ordinal $\alpha$ such that $L_\alpha \models \mathsf{KPi}$, or, on the arithmetical side, such that $L_\alpha \cap \mathscr{P}(\omega)$ is a model of -$\Delta^1_2$-comprehension. (Compare •\ref{CollapseInaccessible}.) +$\Delta^1_2$-comprehension (cf. \cite[theorem VII.3.24 on p. 267 and + theorem VII.5.17 on p. 292 and + errata\footnote{\url{http://www.personal.psu.edu/t20/sosoa/typos.pdf}} + to notes to §VII.5 on p. 293]{Simpson2009}). +(Compare •\ref{CollapseInaccessible}.) This is the smallest ordinal $\omega_1^{\mathsf{E}_1}$ not the order type of a well-ordering recursive in the Tugué |