From 89e97a96a106c846159b31d2236253f0a016b1aa Mon Sep 17 00:00:00 2001
From: "David A. Madore" <david+git@madore.org>
Date: Wed, 20 Feb 2013 15:59:38 +0100
Subject: Fix/clarify what is meant by the various countable collapses.

---
 ordinal-zoo.tex | 34 +++++++++++++++++++++-------------
 1 file changed, 21 insertions(+), 13 deletions(-)

diff --git a/ordinal-zoo.tex b/ordinal-zoo.tex
index e6377a0..43dc6d0 100644
--- a/ordinal-zoo.tex
+++ b/ordinal-zoo.tex
@@ -129,35 +129,43 @@ the Veblen functions with up to that many variables.
 is the proof-theoretic ordinal of Kripke-Platek set
 theory ($\mathsf{KP}$).
 
-\ordinal The collapse of $\Omega_\omega$ (“Takeuti-Feferman-Buchholz
-ordinal”), which is the proof-theoretic ordinal of
-$\Pi^1_1$-comprehension.  [\CHECKTHIS: there may be a confusion
-  between $\Omega_\omega$ and $\Omega_{\omega+1}$ in the collapse.]
+\ordinal The countable collapse of $\varepsilon_{\Omega_\omega + 1}$
+(“Takeuti-Feferman-Buchholz ordinal”), which is the proof-theoretic
+ordinal of $\Pi^1_1$-comprehension + transfinite induction.
 
-\ordinal\label{CollapseInaccessible} The collapse of an inaccessible
+\ordinal\label{CollapseInaccessible} The countable collapse of
+$\varepsilon_{I+1}$ where $I$ is the first inaccessible
 (= $\Pi^1_0$-indescribable) cardinal.  This is the proof-theoretic
 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}.
+arithmetical side, of $\Delta^1_2$-comprehension + transfinite
+induction.  See \cite{JaegerPohlers1983}.
 (Compare •\ref{RecursivelyInaccessible}.)
 
-\ordinal\label{CollapseMahlo} The collapse of a Mahlo cardinal.  This
-is the proof-theoretic ordinal of $\mathsf{KPM}$.
+\ordinal\label{CollapseMahlo} The countable collapse of
+$\varepsilon_{M+1}$ where $M$ is the first Mahlo cardinal.  This is
+the proof-theoretic ordinal of $\mathsf{KPM}$.
 See \cite{Rathjen1990}.  (Compare •\ref{RecursivelyMahlo}.)
 
-\ordinal\label{CollapseWeaklyCompact} The collapse of a weakly compact
+\ordinal\label{CollapseWeaklyCompact} The countable collapse of
+$\varepsilon_{K+1}$ where $K$ is the first weakly compact
 (= $\Pi^1_1$-indescribable) cardinal.  This is the proof-theoretic
 ordinal of $\mathsf{KP} + \Pi_3\hyphen\mathsf{Ref}$.
 See \cite{Rathjen1994}.  (Compare •\ref{RecursivelyWeaklyCompact}.)
 
-\ordinal\label{CollapsePiTwoZeroIndescribable} The collapse of a
+\ordinal\label{CollapsePiTwoZeroIndescribable} The countable collapse
+of $\varepsilon_{\Xi+1}$ where $\Xi$ is the first
 $\Pi^2_0$-indescribable cardinal.  This is the proof-theoretic ordinal
 of $\mathsf{KP} + \Pi_\omega\hyphen\mathsf{Ref}$.
-See \cite[part I]{Stegert2010}.  (Compare •\ref{WeaklyStable}.)
+See \cite[part I]{Stegert2010} (in whose notation this ordinal would
+be called $\Psi^{\varepsilon_{\Xi+1}}_{\mathbb{X}}$ where $\mathbb{X}
+= (\omega^+; \mathsf{P}_0; \epsilon; \epsilon; 0)$).
+(Compare •\ref{WeaklyStable}.)
 
 \ordinal The proof-theoretic ordinal of $\mathsf{Stability}$:
-see \cite[part II]{Stegert2010}.
+see \cite[part II]{Stegert2010} (in whose notation this ordinal would
+be called $\Psi^{\varepsilon_{\Upsilon+1}}_{\mathbb{X}}$ where
+$\mathbb{X} = (\omega^+; \mathsf{P}_0; \epsilon; \epsilon; 0)$).
 
 %
 \section{Recursively large countable ordinals}
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