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| -rw-r--r-- | ordinal-zoo.tex | 25 |
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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 |