From 4ed8f3b926ea09cb21ded12a2bd61686076d115f Mon Sep 17 00:00:00 2001
From: "David A. Madore" <david+git@madore.org>
Date: Thu, 7 Feb 2013 16:38:47 +0100
Subject: Add a note about not including large cardinals.

---
 ordinal-zoo.tex | 18 ++++++++++++++++++
 1 file changed, 18 insertions(+)

(limited to 'ordinal-zoo.tex')

diff --git a/ordinal-zoo.tex b/ordinal-zoo.tex
index 48659ed..e3a506d 100644
--- a/ordinal-zoo.tex
+++ b/ordinal-zoo.tex
@@ -430,6 +430,20 @@ $\Delta^1_2$ (resp. $\Sigma^1_2$) subsets of $\omega$
 This is also the smallest $\Sigma^1_2$-reflecting ordinal
 (\cite{Richter1975}).
 
+\bigbreak
+
+\textbf{\textcolor{orange}{Note:}} This document should probably not
+start listing large cardinals, because \textbf{(0)} the fact that one
+implies the other nonwithstanding, this is about “ordinals”, not
+“cardinals”, \textbf{(1)} they are already well covered elsewhere
+(see, e.g., \cite{Kanamori1997}) and \textbf{(2)} we don't want to
+start making assumptions, e.g., about whether $\omega_1^L$ is or is
+not equal to $\omega_1$, but without making such assumptions it is no
+longer possible to correctly order definitions.  Perhaps a median way
+would be to assume $V=L$ for ordering, forget about measurable
+cardinals and whatnot, and still include inaccessibles, Mahlo, weakly
+compact, etc.
+
 %
 %
 %
@@ -587,6 +601,10 @@ so $A \in L_\gamma$ with $\gamma$ countable, as asserted.
   $R$-operator and the ordinal $\sigma_3$”, \textit{J. Symbolic Logic}
   \textbf{51} (1986), 1–11.
 
+\bibitem[Kanamori1997]{Kanamori1997} Akihiro Kanamori, \textit{The
+  Higher Infinite} (corrected first edition), Perspectives in
+  Mathematical Logic, Springer-Verlag (1997), ISBN 3-540-57071-3.
+
 \bibitem[MarekSrebrny1973]{MarekSrebrny1973} Wiktor Marek \& Marian
   Srebrny, “Gaps in the Constructible Universe”,
   \textit{Ann. Math. Logic} \textbf{6} (1974), 359–394.
-- 
cgit v1.2.3

