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authorDavid A. Madore <david+git@madore.org>2013-01-31 10:36:23 (GMT)
committerDavid A. Madore <david+git@madore.org>2013-01-31 10:36:23 (GMT)
commit41d022555a842d4b78bfb130bd10dd52bc00d9b2 (patch)
treed0c81118cb80179cde9a76fe881c0be3298ce8a0
parent27404099a1b7fbf4316b32c84f2725eefba371e5 (diff)
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More cross-references.
-rw-r--r--ordinal-zoo.tex70
1 files changed, 37 insertions, 33 deletions
diff --git a/ordinal-zoo.tex b/ordinal-zoo.tex
index 9a25dc3..2fef5e0 100644
--- a/ordinal-zoo.tex
+++ b/ordinal-zoo.tex
@@ -28,6 +28,7 @@
\newcommand{\REFTHIS}{\textcolor{brown}{REF THIS}}
\newcommand{\CHECKTHIS}{\textcolor{orange}{CHECK THIS}}
\newcommand{\FIXTHIS}{\textcolor{orange}{FIX THIS}}
+\newcommand{\FINDTHIS}{\textcolor{orange}{FIND THIS}}
%
\newtheorem{ordinalcnt}{Anything}[section]
%\newcounter{ordinalcnt}[section]
@@ -123,20 +124,21 @@ ordinal of Kripke-Platek set theory augmented by the recursive
inaccessibility of the class of ordinals ($\mathsf{KPi}$), or, on the
arithmetical side, of $\Delta^1_2$-comprehension.
See \cite{JaegerPohlers1983}.
+(Compare •\ref{RecursivelyInaccessible}.)
\ordinal\label{CollapseMahlo} The collapse of a Mahlo cardinal. This
is the proof-theoretic ordinal of $\mathsf{KPM}$.
-See \cite{Rathjen1990}.
+See \cite{Rathjen1990}. (Compare •\ref{RecursivelyMahlo}.)
\ordinal\label{CollapseWeaklyCompact} The collapse of a weakly compact
(= $\Pi^1_1$-indescribable) cardinal. This is the proof-theoretic
ordinal of $\mathsf{KP} + \Pi_3-\mathsf{Ref}$.
-See \cite{Rathjen1994}.
+See \cite{Rathjen1994}. (Compare •\ref{RecursivelyWeaklyCompact}.)
\ordinal\label{CollapsePiTwoZeroIndescribable} The collapse of a
$\Pi^2_0$-indescribable cardinal. This is the proof-theoretic ordinal
of $\mathsf{KP} + \Pi_\omega-\mathsf{Ref}$.
-See \cite[part I]{Stegert2010}.
+See \cite[part I]{Stegert2010}. (Compare •\ref{WeaklyStable}.)
\ordinal The proof-theoretic ordinal of $\mathsf{Stability}$:
see \cite[part II]{Stegert2010}.
@@ -160,32 +162,31 @@ admissibles. This ordinal is not admissible. This is the smallest
$\alpha$ such that $L_\alpha \cap \mathscr{P}(\omega)$ is a model of
$\Pi^1_1$-comprehension.
-\ordinal The smallest recursively inaccessible ordinal: this is the
-smallest ordinal which is admissible and limit of admissibles. This
-is the smallest ordinal $\alpha$ such that $L_\alpha \models
-\mathsf{KPi}$, or, on the arithmetical side, such that $L_\alpha \cap
-\mathscr{P}(\omega)$ is a model of $\Delta^1_2$-comprehension.
-(Compare •\ref{CollapseInaccessible}.)
-
-This is the
-smallest ordinal $\omega_1^{\mathsf{E}_1}$ not the order type of a
-well-ordering recursive in the Tugué functional $\mathsf{E}_1$
-(\cite[chapter VIII, theorem 6.6 on p. 421]{Hinman1978}), or
-equivalently, recursive in the hyperjump; and for this $\alpha$ the
-$\alpha$-recursive (resp. $\alpha$-semi-recursive) subsets of $\omega$
-are exactly the the subsets recursive (resp. semi-recursive) in
-$\mathsf{E}_1$ (\cite[chapter VIII, corollary 4.16 on
- p. 412]{Hinman1978}).
+\ordinal\label{RecursivelyInaccessible} The smallest recursively
+inaccessible ordinal: this is the smallest ordinal which is admissible
+and limit of admissibles. This is the smallest ordinal $\alpha$ such
+that $L_\alpha \models \mathsf{KPi}$, or, on the arithmetical side,
+such that $L_\alpha \cap \mathscr{P}(\omega)$ is a model of
+$\Delta^1_2$-comprehension. (Compare •\ref{CollapseInaccessible}.)
+
+This is the smallest ordinal $\omega_1^{\mathsf{E}_1}$ not the order
+type of a well-ordering recursive in the Tugué
+functional $\mathsf{E}_1$ (\cite[chapter VIII, theorem 6.6 on
+ p. 421]{Hinman1978}), or equivalently, recursive in the hyperjump;
+and for this $\alpha$ the $\alpha$-recursive
+(resp. $\alpha$-semi-recursive) subsets of $\omega$ are exactly the
+the subsets recursive (resp. semi-recursive) in $\mathsf{E}_1$
+(\cite[chapter VIII, corollary 4.16 on p. 412]{Hinman1978}).
\ordinal The smallest recursively hyperinaccessible ordinal: i.e., the
smallest recursively inaccessible which is a limit of recursively
inaccessibles.
-\ordinal The smallest recursively Mahlo ordinal: i.e., the smallest
-admissible ordinal $\alpha$ such that for any $\alpha$-recursive
-function $f \colon \alpha \to \alpha$ there is an admissible
-$\beta<\alpha$ which is closed under $f$. This is the smallest
-ordinal $\alpha$ such that $L_\alpha \models \mathsf{KPM}$.
+\ordinal\label{RecursivelyMahlo} The smallest recursively Mahlo
+ordinal: i.e., the smallest admissible ordinal $\alpha$ such that for
+any $\alpha$-recursive function $f \colon \alpha \to \alpha$ there is
+an admissible $\beta<\alpha$ which is closed under $f$. This is the
+smallest ordinal $\alpha$ such that $L_\alpha \models \mathsf{KPM}$.
(Compare •\ref{CollapseMahlo}.)
This is the smallest ordinal not the order type of a well-ordering
@@ -199,16 +200,17 @@ partial normalization of the superjump, \cite[theorem 5 on
Also note concerning this ordinal: \cite[corollary 9.4(ii) on
p. 348]{RichterAczel1974}.
-\ordinal The smallest $\Pi_3$-reflecting (``recursively weakly
-compact'') ordinal. This can also be described as the smallest
-``$2$-admissible'' ordinal: see \cite[theorem 1.16 on
- p. 312]{RichterAczel1974}. (Compare •\ref{CollapseWeaklyCompact}.)
+\ordinal\label{RecursivelyWeaklyCompact} The smallest
+$\Pi_3$-reflecting (``recursively weakly compact'') ordinal. This can
+also be described as the smallest ``$2$-admissible'' ordinal:
+see \cite[theorem 1.16 on p. 312]{RichterAczel1974}.
+(Compare •\ref{CollapseWeaklyCompact}.)
-\ordinal The smallest $(+1)$-stable ordinal, i.e., the smallest
-$\alpha$ such that $L_\alpha \mathrel{\preceq_1} L_{\alpha+1}$. This
-is the smallest $\Pi^1_0$-reflecting (i.e., $\Pi_n$-reflecting for
-every $n\in\omega$) ordinal: \cite[theorem 1.18 on p. 313 and
- 333]{RichterAczel1974}.
+\ordinal\label{WeaklyStable} The smallest $(+1)$-stable ordinal, i.e.,
+the smallest $\alpha$ such that $L_\alpha \mathrel{\preceq_1}
+L_{\alpha+1}$. This is the smallest $\Pi^1_0$-reflecting (i.e.,
+$\Pi_n$-reflecting for every $n\in\omega$) ordinal: \cite[theorem 1.18
+ on p. 313 and 333]{RichterAczel1974}.
(Compare •\ref{CollapsePiTwoZeroIndescribable}.)
@@ -245,6 +247,8 @@ the 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}).
+[\FINDTHIS: how stable is this ordinal?]
+
\ordinal The smallest $(^{++})$-stable ordinal, i.e., the smallest
$\alpha$ such that $L_\alpha \mathrel{\preceq_1} L_{\alpha^{++}}$
where $\alpha^+,\alpha^{++}$ are the two smallest admissible