Reference Avigad's paper for ordinals closed under primitive recursive functions.

1 files changed, 7 insertions, 2 deletions

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