From cac1563244c2993dbef9f23ece3f7d8716188733 Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Wed, 14 Feb 2024 18:44:32 +0100 Subject: [PATCH] silver gist --- .gitea/workflows/build.yaml | 2 +- inputs/lecture_11.tex | 2 +- inputs/lecture_17.tex | 19 +++++++++++++++---- 3 files changed, 17 insertions(+), 6 deletions(-) diff --git a/.gitea/workflows/build.yaml b/.gitea/workflows/build.yaml index 4ba83f4..f1f6861 100644 --- a/.gitea/workflows/build.yaml +++ b/.gitea/workflows/build.yaml @@ -12,7 +12,7 @@ jobs: - name: Prepare pages run: | mkdir public - mv build/logic2.pdf build/logic2.log README.md public + mv build/*.pdf build/*.log README.md public - name: Deploy to pages uses: actions/pages@v1 with: diff --git a/inputs/lecture_11.tex b/inputs/lecture_11.tex index cc55151..d304051 100644 --- a/inputs/lecture_11.tex +++ b/inputs/lecture_11.tex @@ -75,7 +75,7 @@ We often write $\kappa, \lambda, \ldots$ for cardinals. \end{itemize} } \end{proof} -We may now use the \yaref{lem:recursion} +We may now use the \yaref{thm:recursion} to define a sequence $\langle \aleph_\alpha : \alpha \in \OR \rangle$ with the following properties: \begin{IEEEeqnarray*}{rCl} diff --git a/inputs/lecture_17.tex b/inputs/lecture_17.tex index 3cc3123..e7098ff 100644 --- a/inputs/lecture_17.tex +++ b/inputs/lecture_17.tex @@ -236,8 +236,7 @@ We will only proof \item $X \le Y :\iff \{\alpha < \omega_1 : f_X(\alpha) \le f_Y(\alpha)\}$ stationary. \item (2) $X\le Y \lor Y \le X$: Suppose $X \not\le Y, Y \not\le X$. Choose witnessing clubs $C, D$. - $C \cap D$ is club, but then $f_X(\alpha) \le f_Y(\alpha)$ or - $f_X(\alpha) \ge f_Y(\alpha)$ for $\alpha \in C \cap D$. + $C \cap D$ is club, so $C\cap D \ni \alpha \implies f_X(\alpha) \substack{\nleq\\\ngeq} f_Y(\alpha) \lightning$ \item (3) $X \subseteq \aleph_{ \omega_1}$, then $|\underbrace{\{Y \subseteq \aleph_{ \omega_1} : Y \le X\}}_{A}| \le \aleph_{ \omega_1}$ \begin{itemize} \item Suppose $|A| \ge \aleph_{ \omega_1 + 1}$. @@ -245,9 +244,21 @@ We will only proof $2^{\aleph_1} = \aleph_2 \implies$ at most $\aleph_2$ such $S_Y$. \item If $\forall S \in \cP( \omega_1).~ |\underbrace{\{Y \in A : S_Y = S\}}_{A_S}| < \aleph_{ \omega_1 + 1}$, then $|A| \le \aleph_2 \cdot <\aleph_{ \omega_1 + 1}$ $\lightning$ $\aleph_{ \omega_1 + 1}$ regular. - \item So $\exists S \in \omega_1.~|A_S| = \aleph_{ \omega_1 + 1 + 1}$. - % TODO TODO TODO hier weiter! + \item So $\exists S \in \omega_1.~|A_S| = \aleph_{ \omega_1 + 1}$. + \item Fix surjection $\langle g_\alpha : \aleph_\alpha \twoheadrightarrow f_X(\alpha) + 1 : \alpha \in S \rangle$. + ($f_Y(\alpha) \le f_X(\alpha) < \aleph_{\alpha+1}$) + \item $\forall Y \in A_S$ define $\overline{f}_Y \colon S \to \aleph_{ \omega_1}, \alpha \mapsto \min \{\xi : g_\alpha(\xi) = f_Y(\alpha)\}$. + \item $S^\ast$ ($S \cap$ limit ordinals) is stationary. + \item $\forall Y \in A$ define $h_Y\colon S^\ast \to \omega_1, \alpha \mapsto \min \{\beta < \alpha : \overline{f}_Y(\alpha) < \aleph_\beta\}$. + \item Apply \yaref{thm:fodor} to $h_Y, S^\ast$ to get $T_Y \subseteq S^\ast$ stationary with $h_Y\defon{T_Y}$ constant. + \item $\exists T.~|\underbrace{\{Y \in A_S : T_Y = T\}}_{A_{S,T}}| = \aleph_{ \omega_1 + 1}$. + \item Let $\{\beta\} = g_Y''T$, i.e.~$\overline{f}_Y(\alpha) < \aleph_{\beta}$ for $Y\in A_{S,T}, \alpha \in T$. + \item $\leftindex^T \aleph_\beta \le 2^{\aleph_\beta \cdot \aleph_1} = \aleph_{\beta+1} \aleph_2 < \aleph_{ \omega_1}$. + \item $\exists \tilde{f}\colon T \to \aleph_\beta .~ |\underbrace{\{Y \in A_{S,T} : \overline{f}_Y\defon{T} = \tilde{f}\} }_{A_{S,T,\tilde{f}}}| = \aleph_{ \omega_1 + 1}$. + \item $Y,Y' \in A_{S,T,\tilde{f}} \implies \forall \alpha \in T.~f_Y(\alpha) = f_{Y'}\left( \alpha \right) \overset{T \text{ unbounded}}{\implies} Y = Y'$ + Thus $|A_{S,T, \tilde{f}} | \le 1 \lightning$. \end{itemize} + \item Define sequence $\langle X_i : i < \aleph_{ \omega_1 + 1} \rangle$ of subsets of $\aleph_{\omega_1}$: \begin{itemize}