summaryrefslogtreecommitdiffstats
diff options
context:
space:
mode:
authorDavid A. Madore <david+git@madore.org>2013-02-20 14:59:38 (GMT)
committerDavid A. Madore <david+git@madore.org>2013-02-20 15:00:52 (GMT)
commit89e97a96a106c846159b31d2236253f0a016b1aa (patch)
tree649c40a12a09f427fe5015ad9f7ac962f1f31091
parent60b838b3bf3f99a0091edc6b04a5541b5a1fd37d (diff)
downloadordinal-zoo-89e97a96a106c846159b31d2236253f0a016b1aa.zip
ordinal-zoo-89e97a96a106c846159b31d2236253f0a016b1aa.tar.gz
ordinal-zoo-89e97a96a106c846159b31d2236253f0a016b1aa.tar.bz2
Fix/clarify what is meant by the various countable collapses.
-rw-r--r--ordinal-zoo.tex34
1 files 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}