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| -rw-r--r-- | ordinal-zoo.tex | 71 |
1 files changed, 65 insertions, 6 deletions
diff --git a/ordinal-zoo.tex b/ordinal-zoo.tex index 3334068..7e8a784 100644 --- a/ordinal-zoo.tex +++ b/ordinal-zoo.tex @@ -23,6 +23,7 @@ \mathchardef\emdash="07C\relax \mathchardef\hyphen="02D\relax \DeclareUnicodeCharacter{00A0}{~} +\newcommand{\trans}{\mathop{\mathrm{trans}}\nolimits} % % \newcommand{\TODO}{\textcolor{red}{TODO}} @@ -31,10 +32,11 @@ \newcommand{\FIXTHIS}{\textcolor{orange}{FIX THIS}} \newcommand{\FINDTHIS}{\textcolor{orange}{FIND THIS}} % -\newtheorem{ordinalcnt}{Anything}[section] -%\newcounter{ordinalcnt}[section] +\newtheorem{comcnt}{Anything}[section] +%\newcounter{comcnt}[section] \newcommand\ordinal{% -\refstepcounter{ordinalcnt}\medbreak\noindent•\textbf{\theordinalcnt.} } +\refstepcounter{comcnt}\medbreak\noindent•\textbf{\thecomcnt.} } +\newtheorem{prop}[comcnt]{Proposition} % % % @@ -42,7 +44,7 @@ \section{Recursive ordinals} -\setcounter{ordinalcnt}{-1} +\setcounter{comcnt}{-1} \ordinal $0$ (zero). This is the smallest ordinal, and the only one that is neither successor nor limit. @@ -254,8 +256,9 @@ subsets recursive (resp. semi-recursive) in $\mathsf{E}_1^\#$ (combine $\alpha$ such that $L_\alpha \mathrel{\preceq_1} L_{\alpha^{++}}$ where $\alpha^+,\alpha^{++}$ are the two smallest admissible ordinals $>\alpha$. This is $\Sigma^1_1$-reflecting and greater than -the ordinal of •\ref{SigmaOneOne} \cite[theorem 6.4]{Simpson1978} -[\CHECKTHIS, probably similar to \cite[§6]{RichterAczel1974}]. +the ordinal of •\ref{SigmaOneOne} (\cite[theorem 6.4 on + p. 376]{Simpson1978} and +proposition \ref{PlusPlusStableOrdinalIsSigmaOneOneReflecting} below). \ordinal The smallest inaccessibly-stable ordinal, i.e., the smallest $\alpha$ such that $L_\alpha \mathrel{\preceq_1} L_\beta$ where @@ -399,6 +402,54 @@ This is also the smallest $\Sigma^1_2$-reflecting ordinal % % +\section{Various statements} + +\begin{prop}\label{PlusPlusStableOrdinalIsSigmaOneOneReflecting} +If $\alpha$ is such that $L_\alpha \mathrel{\preceq_1} +L_{\alpha^{++}}$ (where $\alpha^+,\alpha^{++}$ are the two smallest +admissible ordinals $>\alpha$) then $\alpha$ is +$\Sigma^1_1$-reflecting. (Stated in \cite[theorem 6.4 on + p. 376]{Simpson1978}.) +\end{prop} +\begin{proof} +Assume $L_\alpha \models \exists U(\varphi(U))$ where $\varphi$ is a +$\Pi^1_0$ (=first-order) formula with constants in $L_\alpha$ and the +extra relation symbol $U$. We want to show that there exists +$\beta<\alpha$ such that $L_\beta \models \exists U(\varphi(U))$. + +Now by \cite[theorem 6.2 on p. 334]{RichterAczel1974} (applied to the +negation of $\exists U(\varphi(U))$) we can find a $\Pi_1$ formula +$\forall z(\psi(S,z))$ (with the same constants as $\varphi$) such +that for any countable transitive set $A$ containing these constants +and any admissible $B\ni A$ we have $B \models \forall z(\theta(A,z))$ +iff $A \models \exists U(\varphi(U))$. + +In particular, $L_{\alpha^+} \models \forall z(\theta(L_\alpha,z))$. +So $L_{\alpha^+} \models \exists A(\trans(A) \land \penalty0 +(A\models\Theta+V{=}L) \land \penalty0 \forall z(\theta(A,z)))$, were +$\Theta$ is a statement which translates the adequacy of $A$ (see +\cite{Jech1978} (13.9) and lemmas 13.2 and 13.3 p. 110–112, or +\cite{Jech2003}, (13.12) and (13.13) p. 188). So in turn +$L_{\alpha^{++}} \models \exists C(\trans(C) \land \penalty0 +(C\models\mathsf{KP}+V{=}L) \land \penalty0 (C \models \exists +A(\trans(A) \land \penalty100 (A\models\Theta+V{=}L) \land \penalty100 +\forall z(\theta(A,z)))))$. But this is a $\Sigma_1$ formula with +constants in $L_\alpha$, so by the assumption we have $L_\alpha +\models$ the same thing. So there is $C \in L_\alpha$ transitive and +containing the constants of $\varphi$, and which is necessarily an +$L_\gamma$ (for $\gamma<\alpha$) because $C \models +\mathsf{KP}+V{=}L$, such that $L_\gamma \models \exists A(\trans(A) +\land \penalty0 (A\models\Theta+V{=}L) \land \penalty0 \forall +z(\theta(A,z)))$. So in turn there exists $A \in L_\gamma$ +transitive, which is necessarily an $L_\beta$ (for $\beta<\gamma$) +because $A \models \Theta+V{=}L$, such that $L_\gamma \models \forall +z(\theta(L_\beta,z))$. So $L_\beta \models \exists U(\varphi(U))$. +\end{proof} + +% +% +% + \begin{thebibliography}{} \bibitem[Aczel1970]{Aczel1970} Peter Aczel, “Representability in Some @@ -448,6 +499,14 @@ This is also the smallest $\Sigma^1_2$-reflecting ordinal Systeme”, \textit{Bayer. Akad. Wiss., Math.-Natur. Kl. Sitzungsber. 1982} (1983), 1–28. +\bibitem[Jech1978]{Jech1978} Thomas Jech, \textit{Set theory}, Pure + and Applied Mathematics \textbf{79}, Academic Press (1978), + ISBN 0-12-381950-4. + +\bibitem[Jech2003]{Jech2003} Thomas Jech, \textit{Set theory, The + third millennium edition, revised and expanded}, Springer Monographs + in Mathematics, Springer-Verlag (2003), ISBN 3-540-44085-2. + \bibitem[Jensen1972]{Jensen1972} Ronald Björn Jensen, “The fine structure of the constructible hierarchy”, \textit{Ann. Math. Logic} \textbf{4} (1972), 229–308. |