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| -rw-r--r-- | ordinal-zoo.tex | 9 |
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 |