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3 changed files with 17 additions and 6 deletions
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@ -12,7 +12,7 @@ jobs:
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- name: Prepare pages
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- name: Prepare pages
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run: |
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run: |
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mkdir public
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mkdir public
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mv build/logic2.pdf build/logic2.log README.md public
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mv build/*.pdf build/*.log README.md public
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- name: Deploy to pages
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- name: Deploy to pages
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uses: actions/pages@v1
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uses: actions/pages@v1
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with:
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with:
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@ -75,7 +75,7 @@ We often write $\kappa, \lambda, \ldots$ for cardinals.
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\end{itemize}
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\end{itemize}
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}
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}
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\end{proof}
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\end{proof}
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We may now use the \yaref{lem:recursion}
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We may now use the \yaref{thm:recursion}
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to define a sequence $\langle \aleph_\alpha : \alpha \in \OR \rangle$
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to define a sequence $\langle \aleph_\alpha : \alpha \in \OR \rangle$
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with the following properties:
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with the following properties:
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\begin{IEEEeqnarray*}{rCl}
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\begin{IEEEeqnarray*}{rCl}
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@ -236,8 +236,7 @@ We will only proof
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\item $X \le Y :\iff \{\alpha < \omega_1 : f_X(\alpha) \le f_Y(\alpha)\}$ stationary.
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\item $X \le Y :\iff \{\alpha < \omega_1 : f_X(\alpha) \le f_Y(\alpha)\}$ stationary.
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\item (2) $X\le Y \lor Y \le X$:
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\item (2) $X\le Y \lor Y \le X$:
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Suppose $X \not\le Y, Y \not\le X$. Choose witnessing clubs $C, D$.
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Suppose $X \not\le Y, Y \not\le X$. Choose witnessing clubs $C, D$.
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$C \cap D$ is club, but then $f_X(\alpha) \le f_Y(\alpha)$ or
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$C \cap D$ is club, so $C\cap D \ni \alpha \implies f_X(\alpha) \substack{\nleq\\\ngeq} f_Y(\alpha) \lightning$
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$f_X(\alpha) \ge f_Y(\alpha)$ for $\alpha \in C \cap D$.
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\item (3) $X \subseteq \aleph_{ \omega_1}$, then $|\underbrace{\{Y \subseteq \aleph_{ \omega_1} : Y \le X\}}_{A}| \le \aleph_{ \omega_1}$
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\item (3) $X \subseteq \aleph_{ \omega_1}$, then $|\underbrace{\{Y \subseteq \aleph_{ \omega_1} : Y \le X\}}_{A}| \le \aleph_{ \omega_1}$
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\begin{itemize}
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\begin{itemize}
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\item Suppose $|A| \ge \aleph_{ \omega_1 + 1}$.
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\item Suppose $|A| \ge \aleph_{ \omega_1 + 1}$.
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@ -245,9 +244,21 @@ We will only proof
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$2^{\aleph_1} = \aleph_2 \implies$ at most $\aleph_2$ such $S_Y$.
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$2^{\aleph_1} = \aleph_2 \implies$ at most $\aleph_2$ such $S_Y$.
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\item If $\forall S \in \cP( \omega_1).~ |\underbrace{\{Y \in A : S_Y = S\}}_{A_S}| < \aleph_{ \omega_1 + 1}$,
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\item If $\forall S \in \cP( \omega_1).~ |\underbrace{\{Y \in A : S_Y = S\}}_{A_S}| < \aleph_{ \omega_1 + 1}$,
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then $|A| \le \aleph_2 \cdot <\aleph_{ \omega_1 + 1}$ $\lightning$ $\aleph_{ \omega_1 + 1}$ regular.
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then $|A| \le \aleph_2 \cdot <\aleph_{ \omega_1 + 1}$ $\lightning$ $\aleph_{ \omega_1 + 1}$ regular.
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\item So $\exists S \in \omega_1.~|A_S| = \aleph_{ \omega_1 + 1 + 1}$.
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\item So $\exists S \in \omega_1.~|A_S| = \aleph_{ \omega_1 + 1}$.
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% TODO TODO TODO hier weiter!
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\item Fix surjection $\langle g_\alpha : \aleph_\alpha \twoheadrightarrow f_X(\alpha) + 1 : \alpha \in S \rangle$.
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($f_Y(\alpha) \le f_X(\alpha) < \aleph_{\alpha+1}$)
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\item $\forall Y \in A_S$ define $\overline{f}_Y \colon S \to \aleph_{ \omega_1}, \alpha \mapsto \min \{\xi : g_\alpha(\xi) = f_Y(\alpha)\}$.
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\item $S^\ast$ ($S \cap$ limit ordinals) is stationary.
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\item $\forall Y \in A$ define $h_Y\colon S^\ast \to \omega_1, \alpha \mapsto \min \{\beta < \alpha : \overline{f}_Y(\alpha) < \aleph_\beta\}$.
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\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.
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\item $\exists T.~|\underbrace{\{Y \in A_S : T_Y = T\}}_{A_{S,T}}| = \aleph_{ \omega_1 + 1}$.
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\item Let $\{\beta\} = g_Y''T$, i.e.~$\overline{f}_Y(\alpha) < \aleph_{\beta}$ for $Y\in A_{S,T}, \alpha \in T$.
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\item $\leftindex^T \aleph_\beta \le 2^{\aleph_\beta \cdot \aleph_1} = \aleph_{\beta+1} \aleph_2 < \aleph_{ \omega_1}$.
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\item $\exists \tilde{f}\colon T \to \aleph_\beta .~ |\underbrace{\{Y \in A_{S,T} : \overline{f}_Y\defon{T} = \tilde{f}\} }_{A_{S,T,\tilde{f}}}| = \aleph_{ \omega_1 + 1}$.
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\item $Y,Y' \in A_{S,T,\tilde{f}} \implies \forall \alpha \in T.~f_Y(\alpha) = f_{Y'}\left( \alpha \right) \overset{T \text{ unbounded}}{\implies} Y = Y'$
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Thus $|A_{S,T, \tilde{f}} | \le 1 \lightning$.
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\end{itemize}
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\end{itemize}
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\item Define sequence $\langle X_i : i < \aleph_{ \omega_1 + 1} \rangle$
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\item Define sequence $\langle X_i : i < \aleph_{ \omega_1 + 1} \rangle$
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of subsets of $\aleph_{\omega_1}$:
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of subsets of $\aleph_{\omega_1}$:
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\begin{itemize}
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\begin{itemize}
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