diff --git a/inputs/lecture_13.tex b/inputs/lecture_13.tex index 7bd0c83..a460e6a 100644 --- a/inputs/lecture_13.tex +++ b/inputs/lecture_13.tex @@ -146,21 +146,21 @@ Relevant concepts to prove this theorem: iff for all $\beta < \alpha$, there is some $\gamma \in \alpha$ such that $\beta < \gamma$. - \item We say that $A \subseteq \alpha$ + \item We say that $A \subseteq \alpha$ is \vocab{closed}, iff it is closed with respect to the order topology on $\alpha$, i.e.~for all $\beta < \alpha$, \[ \sup(A \cap \beta) \in A. - \] - \item $A$ is \vocab{club} (closed unbounded) + \] + \item $A$ is \vocab{club} (\emph{cl}osed \emph{un}bounded) iff it is closed and unbounded. \end{itemize} \end{definition} +The interesting case is that $\alpha$ is a regular uncountable cardinal. \begin{fact} $A \subseteq \alpha$ being unbounded is equivalent to $f\colon \beta\to \alpha$ being cofinal, where $(\beta, \in ) \overset{f}{\cong} (A, \in )$. \end{fact} - diff --git a/inputs/lecture_14.tex b/inputs/lecture_14.tex new file mode 100644 index 0000000..6d0297e --- /dev/null +++ b/inputs/lecture_14.tex @@ -0,0 +1,236 @@ +\lecture{14}{2023-12-04}{} +\begin{abuse} + Sometimes we say club + instead of club in $\kappa$. +\end{abuse} +\begin{example} + Let $\kappa$ be a regular uncountable cardinal. + + \begin{itemize} + \item $\kappa$ is a club in $\kappa$. + \item $\{\xi + 1 : \xi < \kappa\}$ is unbounded in $\kappa$, + but not closed. + \item For each $\alpha < \kappa$, + the set $\alpha + 1 = \{\xi : \xi \le \alpha\}$ + is closed but not unbounded in $\kappa$. + \item $\{\xi < \kappa : \xi \text{ is a limit ordinal}\} $ + is club in $\kappa$. + \end{itemize} +\end{example} +\begin{lemma} + \label{lem:clubintersection} + Let $\kappa$ be regular and uncountable. + Let $\alpha < \kappa$ + and let $\langle C_{\beta} : \beta < \alpha \rangle$ + be a sequence of subsets of $\kappa$ which are all club in $\kappa$. + Then + \[ + \bigcap_{\beta < \alpha} C_{\beta} + \] + is club in $\kappa$. +\end{lemma} +\begin{warning} + This is false for $\alpha = \kappa$: + Let $C_{\beta} \coloneqq \{\xi : \xi > \beta\}$. + Clearly this is club + but $\bigcap_{\beta < \kappa} C_\beta = \emptyset$. + +\end{warning} +\begin{refproof}{lem:clubintersection} + First let $\alpha = 2$. + Let $C, D \subseteq \kappa$ + be a club. + $C \cap D$ is trivially closed: + + Let $\beta < \kappa$. Suppose that $(C \cap D) \cap \beta$ + is unbounded in $\beta$, so $C \cap \beta$ and $D \cap \beta$ + are both unbounded in $\beta$, + so $\beta \in C \cap D$. + + + $C \cap D$ is unbounded: + + Take some $\gamma < \kappa$. + Let $\gamma_0 = \gamma$ + and inductively define $\gamma_n$ : + If $n$ is even, let $\gamma_n \coloneqq \min C \setminus (\gamma_{n-1}+1)$, + otherwise $\gamma_n \coloneqq \min D \setminus (\gamma_{n-1}+1)$. + + Let $\xi = \sup \{\gamma_n : n < \omega\}$. + Then $\xi = \sup \{\gamma_{2n + 2} : n < \omega\} \in D$ + and $\xi \in C$ by the same argument, + so $\xi \in C \cap D$ + (here it is important, that $\cf(\kappa) > \omega$) + and $\xi > \gamma$. + + The case $\alpha > 2$ is similar: + The intersection is closed by exactly the same argument.% + \footnote{``It is even more closed.''} + + Let's prove that $\bigcap \{C_{\beta} : \beta < \alpha\}$ + is unbounded in $\kappa$. + + We will define a sequence $\langle \gamma_i : i \le \alpha \cdot \omega \rangle$% + \footnote{Ordinal multiplication, i.e.~$\alpha \cdot \omega = \sup_{n < \omega} \underbrace{\alpha + \ldots + \alpha}_{n \text{ times}}$.} + as follows: + + Let $\gamma_0 \coloneqq \gamma$. + Choose + \[\gamma_{\alpha \cdot n + \beta + 1} = \min C_{\beta} \setminus (\gamma_{\alpha \cdot n + \beta} + 1)\] + and at limits choose the supremum. + + + Let $\xi = \sup_{i < \alpha \cdot \omega} \gamma_i + = \sup_{i < \omega} \gamma_{\alpha \cdot n + \beta + 1} \in \bigcap_{\beta < \alpha} C_\beta$, + where we have used that. + $\cf(\kappa) > \alpha \cdot \omega$. + +\end{refproof} + +\begin{definition} + $F \subseteq \cP(a)$ is a \vocab{filter} + iff + \begin{enumerate}[(a)] + \item $X,Y \in F \implies X \cap Y \in F$, + \item $X \in F \land X \subseteq Y \subseteq \kappa \implies Y \in F$, + \item $\emptyset \not\in F$, $\kappa \in F$. + \end{enumerate} + + + Let $\alpha \le \kappa$. + We call $F$ \vocab{$< \alpha$-closed} + iff for all $\gamma < \alpha$ and $\{X_\beta : \beta < \gamma\} \subseteq F$ + then $\bigcap \{X_\beta : \beta < \gamma\} \in F$. +\end{definition} +Intuitively, a filter is a collection of ``big'' subsets of $a$. + +\begin{definition} + Let $\kappa$ be regular and uncountable. + The \vocab{club filter} is defined as + \[ + \cF_{\kappa} \coloneqq \{X \subseteq \kappa : \exists \text{ club } C \subseteq \kappa .~ C \subseteq X\}. + \] +\end{definition} +Clearly this is a filter. + +We have shown (assuming \AxC to choose contained clubs): +\begin{theorem} + If $\kappa$ is regular and uncountable. + Then $\cF_\kappa$ is a $< \kappa$-closed filter. +\end{theorem} + +\begin{definition} + Let $\langle A_\beta : \beta < \alpha \rangle$ + be a sequence of sets. + The \vocab{diagonal intersection}, + is defined to be + \[ + + \diagi_{\beta < \alpha} A_{\beta} \coloneqq + \{\xi < \alpha : \xi \in \bigcap \{A_{\beta} : \beta < \xi\} \}. + \] +\end{definition} +\begin{lemma} + Let $\kappa$ be a regular, uncountable cardinal. + If $\langle C_{\beta} : \beta < \kappa \rangle$ + is a sequence of club subsets of $\kappa$, + then $\diagi_{\beta < \kappa} C_{\beta}$ + contains a club. +\end{lemma} +\begin{proof} + Let us fix $\langle C_{\beta} : \beta < \alpha \rangle$. + Write $D_{\beta} \coloneqq \bigcap \{C_{\gamma} : \gamma \le \beta\} $ + for $\beta < \kappa$. + Each $D_{\beta}$ is a club, + $D_{\beta} \subseteq C_{\beta}$ + and $D_{\beta} \supseteq D_{\beta'}$ + for $\beta \le \beta' < \kappa$. + + It suffices to show that $\diagi_{\beta < \kappa} D_{\beta}$ + contains a club. + + \begin{claim} + $\diagi_{\beta < \kappa} D_{\beta}$ is closed in $\kappa$. + \end{claim} + \begin{subproof} + Let $\gamma < \kappa$ be such that $\left( \diagi_{\beta < \kappa} D_{\beta} \right) \cap \gamma$ + is unbounded in $\gamma$. + We aim to show that $\gamma \in \diagi_{\beta < \kappa} D_{\beta}$. + Let $\beta_0 < \gamma$. + We need to see $\gamma \in D_{\beta_0}$. + For each $\beta'$ with $\beta < \beta' < \gamma$, + there is some $\beta'' \in \diagi_{\beta < \kappa} D_\beta$ + with $\beta'' \ge \beta', \beta'' < \gamma$. + In particular $\beta'' \in D_{\beta_0}$. + + We showed that $D_{\beta_0} \cap \gamma$ + is unbounded in $\gamma$, + so $\gamma \in D_{\beta_0}$. + + As $\beta_0 < \gamma$ was arbitrary, + this shows that $\gamma \in \diagi_{\beta < n} D_\beta$. + \end{subproof} + + \begin{claim} + $\diagi_{\beta < \kappa} D_{\beta}$ + is unbounded in $\kappa$. + \end{claim} + \begin{subproof} + Fix $\gamma < \kappa$. + We need to find $\delta > \gamma$ + with $\gamma \in \diagi_{\beta < \kappa} D_\beta$. + + Define $\langle \gamma_n : n < \omega \rangle$ + as follows: + $\gamma_0 \coloneqq \gamma$ + and + \[ + \gamma_{n+1} \coloneqq \min D_{\gamma_n} \setminus (\gamma_n + 1) + \] + + We have $\delta \coloneqq \sup_{n < \omega} \gamma_n \in \kappa$ + by cofinality of $\kappa$. + + We need to show that $\delta \in D_{\overline{\gamma}}$ + for all $\overline{\gamma} < \delta$. + + If $\overline{\gamma} < \delta$, then $\overline{\gamma} \le \gamma_n$ + for some $n < \omega$. + For $m \ge n$, $\gamma_{m+1} \in D_{\gamma_m} \subseteq D_{\gamma_n} \subseteq D_{\overline{\gamma}}$. + So $D_{\overline{\gamma}} \cap \delta$ is unbounded + in $\gamma$, hence $\delta \in D_{\overline{\gamma}}$. + \end{subproof} +\end{proof} + +\begin{definition} + Let $\kappa$ be regular and uncountable. + $S \subseteq \kappa$ is called \vocab{stationary} (in $\kappa$) + iff $C \cap S \neq \emptyset$ + for every club $C \subseteq \kappa$. +\end{definition} +\begin{example} + \begin{itemize} + \item Every $D \subseteq \kappa$ which is club in $\kappa$ + is stationary in $\kappa$. + \item There exist disjoint stationary sets:\footnote{Note that clubs can never be disjoint, since their intersection is a club.} + Let $\kappa = \omega_2$. + Let $S_0 \coloneqq \{\xi < \kappa : \cf(\xi) = \omega\}$ + and $S_1 \coloneqq \{\xi < \kappa : \cf(\xi) = \omega_1\}$. + Clearly these are disjoint. + They are both stationary: + Let $c \subseteq \kappa$ be a club. + Let $(\xi_i : i \le \omega_1)$ + be defined as follows: + $\xi_0 \coloneqq \min C$, + $\xi_i \coloneqq \min (C \setminus \sup_{j < i} \xi_j)$. + For $i \le \omega_1$ we have that $\xi_i = \sup_{j < i} \xi_j$. + In particular $\xi_\omega \in S_0 \cap C$ + and $\xi_{\omega_1} \in S_1 \cap C$. + \end{itemize} +\end{example} +We will show later that if $ \kappa$ is a regular uncountable cardinal, +then every stationary $S \subseteq \kappa$ can be written as +$S = \bigcup_{i < \kappa} S_i$, +where the $S_i$ are stationary and pairwise disjoint. + + diff --git a/logic.sty b/logic.sty index 21839af..5f0f75a 100644 --- a/logic.sty +++ b/logic.sty @@ -142,4 +142,6 @@ \DeclareSimpleMathOperator{cf} \newcommand\lecture[3]{\hrule{\color{darkgray}\hfill{\tiny[Lecture #1, #2]}}} +%\newcommand\diagi{\mathop{\large \Delta}\limits} +\newcommand\diagi{\mathop{\large \Delta}} diff --git a/logic2.tex b/logic2.tex index 70d8614..1e132f3 100644 --- a/logic2.tex +++ b/logic2.tex @@ -37,6 +37,7 @@ \input{inputs/lecture_11} \input{inputs/lecture_12} \input{inputs/lecture_13} +\input{inputs/lecture_14} \cleardoublepage