path: root/ordinal-zoo.tex
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authorDavid A. Madore <>2017-07-29 21:42:18 +0200
committerDavid A. Madore <>2017-07-29 21:42:18 +0200
commitf9d3012c3a6eff841ab6ab6a74788b31d90b84ec (patch)
treed21964ed09736c310bb71c330afd8cf1723c9a47 /ordinal-zoo.tex
parent4644226a5cfbffc55fa987cb54b6d990f4d0d77e (diff)
Add some more references to Simpson's SoSOA.
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1 files changed, 8 insertions, 2 deletions
diff --git a/ordinal-zoo.tex b/ordinal-zoo.tex
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@@ -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 (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{}}
+ 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