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diff --git a/ordinal-zoo.tex b/ordinal-zoo.tex index e3a506d..540f8ef 100644 --- a/ordinal-zoo.tex +++ b/ordinal-zoo.tex @@ -162,11 +162,11 @@ see \cite[part II]{Stegert2010}. % \section{Recursively large countable ordinals} -\ordinal The Church-Kleene ordinal $\omega_1^{\mathrm{CK}}$: the -smallest admissible ordinal $>\omega$. This is the smallest ordinal -which is not the order type of a recursive (equivalently: -hyperarithmetic) well-ordering on $\omega$. The -$\omega_1^{\mathrm{CK}}$-recursive +\ordinal\label{ChurchKleene} The Church-Kleene ordinal +$\omega_1^{\mathrm{CK}}$: the smallest admissible ordinal $>\omega$. +This is the smallest ordinal which is not the order type of a +recursive (equivalently: hyperarithmetic) well-ordering on $\omega$. +The $\omega_1^{\mathrm{CK}}$-recursive (resp. $\omega_1^{\mathrm{CK}}$-semi-recursive) subsets of $\omega$ are exactly the $\Delta^1_1$ (=hyperarithmetic) (resp. $\Pi^1_1$) subsets of $\omega$, and they are also exactly the subsets recursive @@ -194,6 +194,15 @@ and for this $\alpha$ the $\alpha$-recursive subsets recursive (resp. semi-recursive) in $\mathsf{E}_1$ (\cite[chapter VIII, corollary 4.16 on p. 412]{Hinman1978}). +This is the smallest $\alpha$ such that $L_\alpha \models +\mathsf{KP}+\mathit{Beta}$, where $\mathit{Beta}$ asserts the +existence of a transitive collapse for any well-founded relation, or +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 The smallest recursively hyperinaccessible ordinal: i.e., the smallest recursively inaccessible which is a limit of recursively inaccessibles. @@ -574,6 +583,10 @@ so $A \in L_\gamma$ with $\gamma$ countable, as asserted. $R$-operator and the first nonprojectible ordinal”, unpublished notes (1975). +\bibitem[Harrison1968]{Harrison1968} Joseph Harrison, “Recursive + pseudo-well-orderings”, \textit{Trans. Amer. Math. Soc.} + \textbf{131} (1968), 526–543. + \bibitem[Hinman1978]{Hinman1978} Peter G. Hinman, \textit{Recursion-Theoretic Hierarchies}, Perspectives in Mathematical Logic \textbf{9}, Springer-Verlag (1978), @@ -609,6 +622,10 @@ so $A \in L_\gamma$ with $\gamma$ countable, as asserted. Srebrny, “Gaps in the Constructible Universe”, \textit{Ann. Math. Logic} \textbf{6} (1974), 359–394. +\bibitem[Nadel1973]{Nadel1973} Mark Nadel, “Scott Sentences and + Admissible Sets”, \textit{Ann. Math. Logic} \textbf{7} (1974), + 267–294. + \bibitem[Putnam1963]{Putnam1963} Hilary Putnam, “A Note on Constructible Sets of Integers”, \textit{Notre Dame J. Formal Logic} \textbf{4} (1963), 270–273. |