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authorDavid A. Madore <david+git@madore.org>2013-02-07 16:15:47 (GMT)
committerDavid A. Madore <david+git@madore.org>2013-02-07 16:15:47 (GMT)
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Note on pseudo-well-orderings.
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@@ -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$R$-operator and the first nonprojectible ordinal”, unpublished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.