From cc63a0d62cc846c49a0ad4f0991f9debedfcbba8 Mon Sep 17 00:00:00 2001
From: "David A. Madore" <david+git@madore.org>
Date: Wed, 6 Feb 2013 19:29:55 +0100
Subject: More on reflecting ordinals and the semirecursive subsets they
 define.

---
 ordinal-zoo.tex | 25 +++++++++++++++++++++----
 1 file changed, 21 insertions(+), 4 deletions(-)

(limited to 'ordinal-zoo.tex')

diff --git a/ordinal-zoo.tex b/ordinal-zoo.tex
index 7e8a784..996acc5 100644
--- a/ordinal-zoo.tex
+++ b/ordinal-zoo.tex
@@ -209,6 +209,13 @@ also be described as the smallest ``$2$-admissible'' ordinal:
 see \cite[theorem 1.16 on p. 312]{RichterAczel1974}.
 (Compare •\ref{CollapseWeaklyCompact}.)
 
+Also the sup of the closure ordinals for $\Sigma_3$ inductive
+operators: \cite[theorem A on p. 303]{RichterAczel1974}.  For this
+$\alpha$ the $\alpha$-semi-recursive subsets of $\omega$ are exactly
+the $\Sigma_3$-inductively definable subsets of $\omega$
+(\cite[theorem A on p. 303 and theorem D on p. 304]{RichterAczel1974};
+see also \cite[example 4.12 on p. 370]{Simpson1978}).
+
 \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.,
@@ -224,7 +231,11 @@ ordinal $>\alpha$.  This is the smallest $\Pi^1_1$-reflecting ordinal:
 \cite[theorem 1.19 on p. 313 and 336]{RichterAczel1974}.  Also the sup
 of the closure ordinals for $\Pi^1_1$ inductive operators:
 \cite[theorem B on p. 303 or 10.7 on p. 355]{RichterAczel1974} and
-\cite[theorem A on p. 222]{Cenzer1974}.
+\cite[theorem A on p. 222]{Cenzer1974}.  For this $\alpha$ the
+$\alpha$-semi-recursive subsets of $\omega$ are exactly the
+$\Pi^1_1$-inductively definable subsets of $\omega$ (\cite[theorem D
+  on p. 304]{RichterAczel1974}; see also \cite[example 4.13 on
+  p. 370]{Simpson1978}).
 
 This is the smallest ordinal $\omega_1^{\mathsf{G}_1^\#}$ not the
 order type of a well-ordering recursive in the nondeterministic
@@ -237,9 +248,15 @@ $\mathsf{G}_1^\#$ (\cite[theorem 7.4 on p. 238]{Cenzer1974}).
 \ordinal\label{SigmaOneOne} The smallest $\Sigma^1_1$-reflecting
 ordinal.  Also the sup of the closure ordinals for $\Sigma^1_1$
 inductive operators: \cite[theorem B on p. 303 or 10.7 on
-  p. 355]{RichterAczel1974}.  That this ordinal is smaller than that
-of •\ref{PiOneOne}: \cite[corollary 1 to theorem 6 on
-  p.213]{Anderaa1974}; also see: \cite[theorem 6.5]{Simpson1978}.
+  p. 355]{RichterAczel1974}.  For this $\alpha$ the
+$\alpha$-semi-recursive subsets of $\omega$ are exactly the
+$\Sigma^1_1$-inductively definable subsets of $\omega$
+(\cite[theorem D on p. 304]{RichterAczel1974}; see
+also \cite[example 4.14 on p. 370]{Simpson1978}).
+
+That this ordinal is gerater than that of •\ref{PiOneOne}:
+\cite[corollary 1 to theorem 6 on p.213]{Anderaa1974}; also see:
+\cite[theorem 6.5]{Simpson1978}.
 
 This is the smallest ordinal $\omega_1^{\mathsf{E}_1^\#}$ not the
 order type of a well-ordering recursive in the nondeterministic
-- 
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