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author committer David A. Madore 2013-02-20 15:19:27 (GMT) David A. Madore 2013-02-20 15:19:27 (GMT) 741c8fc0752470f7314a2b39ca7bfbd38e9e59e3 (patch) 0de34bc203d4d1c06165c993c0209820a9683e54 89e97a96a106c846159b31d2236253f0a016b1aa (diff) ordinal-zoo-741c8fc0752470f7314a2b39ca7bfbd38e9e59e3.zipordinal-zoo-741c8fc0752470f7314a2b39ca7bfbd38e9e59e3.tar.gzordinal-zoo-741c8fc0752470f7314a2b39ca7bfbd38e9e59e3.tar.bz2
Reference Avigad's paper for ordinals closed under primitive recursive functions.
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1 files changed, 7 insertions, 2 deletions
 diff --git a/ordinal-zoo.tex b/ordinal-zoo.texindex 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