Give proof of the fact that (++)-stable ordinals are Σ¹₁-reflecting.

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.