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authorDavid A. Madore <david+git@madore.org>2013-02-20 15:19:27 (GMT)
committerDavid A. Madore <david+git@madore.org>2013-02-20 15:19:27 (GMT)
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Reference Avigad's paper for ordinals closed under primitive recursive functions.
-rw-r--r--ordinal-zoo.tex9
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diff --git a/ordinal-zoo.tex b/ordinal-zoo.tex
index 43dc6d0..5e796a1 100644
--- a/ordinal-zoo.tex
+++ b/ordinal-zoo.tex
@@ -100,8 +100,9 @@ proof-theoretic ordinal of Peano arithmetic.
\varphi(\gamma+1,\alpha)$ is defined as the function enumerating the
fixed points of $\xi \mapsto \varphi(\gamma,\xi)$.
-\ordinal $\varphi(\omega,0)$. This is the smallest ordinal closed
-under primitive recursive ordinal functions.
+\ordinal $\varphi(\omega,0)$. This is the smallest ordinal $>\omega$
+closed under primitive recursive ordinal functions
+(\cite[corollary 4.5]{Avigad2002}).
\ordinal The Feferman-Schütte ordinal $\Gamma_0 = \varphi(1,0,0)$
(also $\psi(\Omega^{\Omega})$ for an appropriate collapsing
@@ -578,6 +579,10 @@ so $A \in L_\gamma$ with $\gamma$ countable, as asserted.
\bibitem[Adams1981]{Adams1981} Douglas Adams, \textit{The Hitchiker's
Guide to the Galaxy}, Pocket Books (1981), ISBN 0-671-46149-4.
+\bibitem[Avigad2002]{Avigad2002} Jeremy Avigad, “An ordinal analysis
+ of admissible set theory using recursion on ordinal notations”,
+ \textit{J. Math. Log.} \textbf{2} (2002), 91–112.
+
\bibitem[Barwise1975]{Barwise1975} Jon Barwise, \textit{Admissible
sets and structures, An approach to definability theory},
Perspectives in Mathematical Logic \textbf{7}, Springer-Verlag