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authorDavid A. Madore <david+git@madore.org>2017-07-29 19:28:22 (GMT)
committerDavid A. Madore <david+git@madore.org>2017-07-29 19:28:22 (GMT)
commit4644226a5cfbffc55fa987cb54b6d990f4d0d77e (patch)
treed5daa9118de9f0ae70032135e75ac17f693c68cc
parent18276c8a7f480670b9582dbd1a6387725df1421b (diff)
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Improve some references.
-rw-r--r--ordinal-zoo.tex13
1 files changed, 9 insertions, 4 deletions
diff --git a/ordinal-zoo.tex b/ordinal-zoo.tex
index 81c37ca..54b4f82 100644
--- a/ordinal-zoo.tex
+++ b/ordinal-zoo.tex
@@ -182,7 +182,9 @@ subsets of $\omega$, and they are also exactly the subsets recursive
[this is stated vaguely and without proof in \cite[§2, introductory
remarks]{HinmanMoschovakis1971}, and also alluded to, but with an
argument, in \cite[chapter VI, introductory remarks to §6 on
- p. 316]{Hinman1978}]).
+ p. 316]{Hinman1978}; but the essential argument should be Gandy's
+ selection theorem, \cite[chapter VI, theorem 4.1 on
+ p. 292 or its corollary 4.3 on p. 294]{Hinman1978}]).
\ordinal $\omega_\omega^{\mathrm{CK}}$: the smallest limit of
admissibles. This ordinal is not admissible. This is the smallest
@@ -212,7 +214,8 @@ equivalently, the smallest admissible $\alpha$ such that any ordering
which $L_\alpha$ thinks is a well-ordering is, indeed, a
well-ordering: see \cite[theorem 6.1 on p. 291]{Nadel1973}
(compare \cite{Harrison1968} for the negative result concerning the
-ordinal $\omega_1^{\mathrm{CK}}$ of •\ref{ChurchKleene}).
+ordinal $\omega_1^{\mathrm{CK}}$ of •\ref{ChurchKleene}; compare also
+\cite{Gostanian1979} and •\ref{SigmaOneOne} for related facts).
\ordinal The smallest recursively hyperinaccessible ordinal: i.e., the
smallest recursively inaccessible which is a limit of recursively
@@ -304,8 +307,10 @@ This is the smallest admissible $\alpha$ which is not Gandy, i.e.,
such that every $\alpha$-recursive linear ordering of $\alpha$ of
which $L_{\alpha^+}$ thinks that it is a well-ordering (with
$\alpha^+$ being the next admissible) is, indeed, a well-ordering: see
-\cite[theorem 6.6 on p. 377]{Simpson1978}, \cite[\FINDTHIS:
- where ?]{AbramsonSacks1976} and \cite[theorem 3.3]{Gostanian1979}.
+\cite[theorem 6.6 on p. 377]{Simpson1978} and
+\cite[theorem 3.3]{Gostanian1979} (on the terminology ``Gandy
+ordinal'', see \cite{AbramsonSacks1976}: in \cite{Gostanian1979} the
+same ordinals are called ``good'').
[\FINDTHIS: how stable is this ordinal?]