silver gist

.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:

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}

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}