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authorDavid A. Madore <david+git@madore.org>2013-02-07 16:40:40 (GMT)
committerDavid A. Madore <david+git@madore.org>2013-02-07 16:40:40 (GMT)
commit72ac39542e5c463de5751c1392cf17220334c02b (patch)
treec73ea273c80143b1dc0bdefd14f086edfe3a38cb
parent53ed97c1dc23de71d469acb596d68058df410b00 (diff)
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Note on Gandy ordinals.
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@@ -289,6 +289,13 @@ subsets recursive (resp. semi-recursive) in $\mathsf{E}_1^\#$ (combine
\cite[theorem 1 on p. 313, theorem 2 on p. 318]{Aczel1970} and
\cite[theorem D on p. 304]{RichterAczel1974}).
+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} and \cite[\FINDTHIS:
+ where ?]{AbramsonSacks1976}.
+
[\FINDTHIS: how stable is this ordinal?]
\ordinal The smallest $(^{++})$-stable ordinal, i.e., the smallest
@@ -540,6 +547,10 @@ so $A \in L_\gamma$ with $\gamma$ countable, as asserted.
\begin{thebibliography}{}
+\bibitem[AbramsonSacks1976]{AbramsonSacks1976} Fred G. Abramson \&
+ Gerald E. Sacks, “Uncountable Gandy Ordinals”, \textit{J. London
+ Math. Soc. (2)} \textbf{14} (1976), 387–392.
+
\bibitem[Aczel1970]{Aczel1970} Peter Aczel, “Representability in Some
Systems of Second Order Arithmetic”, \textit{Israel J. Math}
\textbf{8} (1970), 308–328.