From 41d022555a842d4b78bfb130bd10dd52bc00d9b2 Mon Sep 17 00:00:00 2001
From: "David A. Madore" <david+git@madore.org>
Date: Thu, 31 Jan 2013 11:36:23 +0100
Subject: More cross-references.

---
 ordinal-zoo.tex | 70 ++++++++++++++++++++++++++++++---------------------------
 1 file changed, 37 insertions(+), 33 deletions(-)

(limited to 'ordinal-zoo.tex')

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
-- 
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