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diff --git a/ordinal-zoo.tex b/ordinal-zoo.tex index 540f8ef..54cdc13 100644 --- a/ordinal-zoo.tex +++ b/ordinal-zoo.tex @@ -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. |