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| -rw-r--r-- | ordinal-zoo.tex | 13 |
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?] |