Merge branch 'main' of https://git.abstractnonsen.se/josia-notes/w23-logic-3

inputs/lecture_24.tex

@@ -0,0 +1,178 @@
+\lecture{24}{2024-01-23}{Combinatorics!}
+
+
+\subsection{Applications to Combinatorics} % Ramsey Theory}
+
+% TODO Define Ultrafilter
+
+\begin{definition}
+    An \vocab{ultrafilter} on $\N$ (or any other set)
+    is a family $\cU \subseteq \cP(\N)$
+    such that
+    \begin{enumerate}[(1)]
+        \item $X \in \cU \land X \subseteq Y \subseteq \N \implies Y \in \cU$.
+        \item $X,Y \in \cU \implies X \cap Y \in \cU$.
+        \item $\emptyset \not\in \cU$, $\N \in \cU$.
+        \item For all $X \subseteq \N$,
+            we have $X \in \cU \lor \N \setminus X \in \cU$.
+    \end{enumerate}
+\end{definition}
+\begin{remark}
+    \begin{itemize}
+        \item If $X \cup Y \in \cU$ then $X \in \cU \lor Y$ or $Y \in \cU$:
+            Consider $((\N \setminus X) \cap (\N \setminus Y) = \N  \setminus (X \cup Y)$.
+        \item Every filter can be extended to an ultrafilter.
+            (Zorn's lemma)
+    \end{itemize}
+\end{remark}
+\begin{definition}
+    An ultrafilter is called \vocab[Ultrafilter!principal]{principal} or \vocab[Ultrafilter!trivial]{trivial} 
+    if it is of the form
+    \[
+    \hat{n} = \{X \subseteq \N : n \in X\}.
+    \]
+\end{definition}
+\begin{notation}
+    Let $\phi(\cdot )$ be a formula, where the argument is a natural number.
+    Let $\cU$ be an ultrafilter.
+    We write
+    \[
+        (\cU n) ~ \phi(n)
+    \]
+    for $\{ n \in \N : \phi(n)\} \in \cU$.
+    We say that $\phi(n)$ holds for  \vocab{$\cU$-almost all}  $n$.
+\end{notation}
+\begin{observe}
+    Let $\phi(\cdot )$, $\psi(\cdot )$ be formulas.
+
+    \begin{enumerate}[(1)]
+        \item $(\cU n) (\phi(n) \land \psi(m)) \iff (\cU n) \phi(n) \land (\cU n) \psi(n)$.
+        \item $(\cU n) (\phi(n) \lor \psi(m)) \iff (\cU n) \phi(n) \lor (\cU n) \psi(n)$.
+        \item $(\cU n) \lnot \phi(n) \iff \lnot (\cU n) \phi(n)$.
+    \end{enumerate}
+\end{observe}
+\begin{lemma}
+    \label{lem:ultrafilterlimit}
+    Let $X $ be a compact Hausdorff space.
+    Let $\cU$ be an ultrafilter.
+    Then for every sequence $(x_n)$ in $X$,
+    there is a unique $x \in X$,
+    such that
+    \[
+    (\cU_n) (x_n \in G)
+    \]
+    for every neighbourhood%
+    \footnote{$G \subseteq X$ is a neighbourhood iff $x \in \inter G$.}
+    $G$ of $x$.
+\end{lemma}
+\begin{notation}
+    In this case we write $x = \cU-\lim_n x_n$.
+\end{notation}
+\begin{refproof}{lem:ultrafilterlimit}[sketch]
+    Whenever we write $X = Y \cup Z$
+    we have $(\cU n) x_n \in Y$
+    or $(\cU n) x_n \in Z$.
+
+    So we can repeatedly chop the space in two pieces,
+    one of them contains $\cU$-almost all $x_n$,
+    Then we restrict to this piece and continue.
+
+    For this to work, we need
+    a finite collection $\cP_n$ of closed sets for every $n$,
+    such that $\bigcup \cP_n = X$,
+    $C \in \cP_{n+1} \implies \exists  C \subseteq D \in \cP_{n}$
+    and
+     $C_1 \supseteq C_2 \supseteq \ldots$, $C_i \in \cP_i $ $\implies | \bigcap_{i} C_i| = 1$.
+    It is clear that we can do this for metric spaces,
+    but such partition can be found for compact Hausdorff spaces as well.
+\end{refproof}
+
+Let $\beta \N$ be the Čech-Stone compactification of $\N$,
+i.e.~the set of all ultrafilters on $\N$
+with the topology given by open sets $V_{A} = \{ p \in \beta\N : A \in P\} $
+for $A \subseteq  \N$.
+
+This is a compact Hausdorff space.\todo{Homework}
+We can turn it into a compact semigroup:
+Consider $+ \colon \N \times  \N \to  \N$.
+This gives an operation on principal ultrafilters
+(we identify $n \in \N$ with the corresponding principal filter).
+We want to extend this to all of $\beta\N$.
+Fix the first argument to get a function $\N \to  \N, n \mapsto k+n$.
+For $\cU \in \beta\N$ consider $\cU-\lim_n (k+n)$.
+So for a fixed $k \in \N$ we get $k+ \cdot \colon\beta\N \to  \beta\N$,
+i.e.~$+ \colon \N \times  \beta\N \to  \beta\N$.
+Fixing the second coordinate to be $\cV \in \beta\N$,
+we get a function $+\cV \colon \N \to \beta\N$.
+For $ \cU \in \beta\N$
+consider $\cU-\lim_n n + \cV$.
+This gives $+ \colon \beta\N \times  \beta\N \to  \beta\N$.
+% TODO ?
+
+\[
+\cU + \cV = \{X \subseteq \N : \{m \colon  \{n \colon  m+n \in X\} \in \cV \} \in \cU \}.
+\]
+
+
+This is not commutative,
+but associative and $a \mapsto a + b$ is continuous
+for a fixed  $b$.
+This is called a left compact topological semigroup.
+
+
+
+
+Let $X$ be a compact Hausdorff space
+and let $T \colon  X \to  X$ be continuous.%
+\footnote{Note that this need not be a homeomorphism, i.e.~we only get a $\N$-action
+    but not a $\Z$-action.}
+
+For any $\cU \in \beta\N$, we define $T^{\cU}$ by
+$T^\cU(x) \coloneqq  \cU-\lim_n T^n(x)$ for $x \in X$.
+
+For fixed $x$, the map $\cU \mapsto T^{\cU}(x)$ is continuous.
+
+(More generally, for every $f\colon  \N \to  X$
+the extension $\tilde{f}\colon \beta\N \to X$ is continuous).
+
+Note that for fixed $\cU$, the map $x \mapsto T^\cU(x)$
+is not necessarily continuous.
+
+
+\begin{definition}
+    Let $X$ be a compact Hausdorff space
+    and $T\colon X\to X$ continuous.
+    A point $x \in X$ is \vocab{recurrent},
+    iff for every neighbourhood $G$ of $x$,
+    infinitely many $n$ satisfy $T^n(x) \in G$.
+
+    A point $x \in X$ is \vocab{uniformly recurrent},
+    if for every neighbourhood $G$ of $x$,
+    there exists $M \in \N$,
+    such that
+    \[
+    \forall n.~\exists k < M.~ T^{n+k}(x) \in G.
+    \]
+
+\end{definition}
+\begin{fact}
+    Let $\cU, \cV \in \beta\N$
+    and $T\colon  X \to  X$ continuous
+    for a compact Hausdorff space $X$.
+    Then $T^{\cU}(T^{\cV}(x)) = T^{\cU + \cV}(x)$.
+\end{fact}
+\begin{proof}
+    \begin{IEEEeqnarray*}{rCl}
+        T^{\cU + \cV}(x) &=& (\cU + \cV)-\lim_k T^k(x)\\
+                         &=& \cU-\lim_m \cV-\lim_n T^{m+n}(x)\\
+                         &\overset{T^m \text{ continuous}}{=}& \cU-\lim_m T^m (\cV-\lim_n T^n(x))\\
+                         &=& T^\cU(T^\cV(x)).
+    \end{IEEEeqnarray*}
+\end{proof}
+
+
+\todo{Homework: Check the details that were omitted during the lecture.}
+
+
+
+

logic3.tex

@@ -50,6 +50,7 @@
 \input{inputs/lecture_21}
 \input{inputs/lecture_22}
 \input{inputs/lecture_23}
+\input{inputs/lecture_24}
 
 \cleardoublepage