From 515ff64ac6cc89f929a9f71948a87454b363bf33 Mon Sep 17 00:00:00 2001
From: Josia Pietsch <git@jrpie.de>
Date: Wed, 14 Feb 2024 15:54:27 +0100
Subject: [PATCH] some small changes

---
 inputs/lecture_16.tex |  1 +
 inputs/lecture_18.tex | 13 +++++++------
 2 files changed, 8 insertions(+), 6 deletions(-)

diff --git a/inputs/lecture_16.tex b/inputs/lecture_16.tex
index 275f4b9..d919e49 100644
--- a/inputs/lecture_16.tex
+++ b/inputs/lecture_16.tex
@@ -201,6 +201,7 @@ one cofinality.
         \label{thm:solovay:p:c2}
         Each $T_i$ is stationary
         and if $i \neq j$, then $T_i \cap T_j = \emptyset$.
+        \footnote{maybe this should not be a claim}
     \end{claim}
     \begin{subproof}
         The first part is true by construction.
diff --git a/inputs/lecture_18.tex b/inputs/lecture_18.tex
index 0320872..2d99308 100644
--- a/inputs/lecture_18.tex
+++ b/inputs/lecture_18.tex
@@ -42,13 +42,14 @@
 \begin{definition}[Ulam]
     A cardinal $\kappa > \aleph_0$ is \vocab{measurable} 
     iff there is an ultrafilter $U$ on $\kappa$,
-    such that $U$ is not principal\footnote{%
+    such that $U$ is not principal\gist{\footnote{%
         i.e.~$\{\xi\}  \not\in U$ for all $\xi < \kappa$%
-    }
-    and
-    if $\theta < \kappa$
+    }}{}
+    and $< \kappa$-closed\gist{,%
+    i.e.~if $\theta < \kappa$
     and $\{X_i : i < \theta\} \subseteq  U$,
-    then $\bigcap_{i < \theta} X_i \in U$
+    then $\bigcap_{i < \theta} X_i \in U$.
+    }{.}
 \end{definition}
 
 \begin{goal}
@@ -77,7 +78,7 @@
 \end{theorem}
 \begin{proof}
     2. $\implies$ 1.:
-    Fox $j\colon  V \to M$.
+    Fix $j\colon  V \to M$.
     Let $U = \{X \subseteq  \kappa : \kappa \in j(X)\}$.
     We need to show that $U$ is an ultrafilter:
     \begin{itemize}