From 26c32ce1bb94cf6b77b029c24cf23bf3047cc7d3 Mon Sep 17 00:00:00 2001
From: Josia Pietsch <git@jrpie.de>
Date: Mon, 13 Nov 2023 20:28:21 +0100
Subject: [PATCH] small fixes

---
 inputs/lecture_04.tex | 2 +-
 inputs/lecture_06.tex | 6 +++---
 inputs/lecture_07.tex | 9 ---------
 inputs/lecture_08.tex | 2 +-
 4 files changed, 5 insertions(+), 14 deletions(-)

diff --git a/inputs/lecture_04.tex b/inputs/lecture_04.tex
index 6f12fa5..fd5bf47 100644
--- a/inputs/lecture_04.tex
+++ b/inputs/lecture_04.tex
@@ -10,7 +10,7 @@ $\ZFC$ stands for
 \begin{itemize}
     \item \textsc{Zermelo}’s axioms (1905), % crises around 19000
     \item \textsc{Fraenkel}'s axioms,
-    \item the axiom of choice.
+    \item the \yaref{ax:c}.
 \end{itemize}
 \begin{notation}
     We write $x \subseteq y$ as a shorthand
diff --git a/inputs/lecture_06.tex b/inputs/lecture_06.tex
index 5b68cd6..98ceb3b 100644
--- a/inputs/lecture_06.tex
+++ b/inputs/lecture_06.tex
@@ -14,7 +14,7 @@
     Note further that if $b_1 \neq  b_2$,
     then $\{(b_1, x) : x \in  b_1\} $
     and $\{(b_2, x) : x \in b_2\}$ are disjoint.
-    Hence the axiom of choice
+    Hence the \yaref{ax:c}
     gives us a choice function $f$ on $A$,
     i.e.~$\forall b \in  \cP(a) \setminus \{\emptyset\} .~(f(b) \in b)$.
 
@@ -68,7 +68,7 @@
 \end{refproof}
 
 \begin{remark}
-    Over $\ZF$ the axiom of choice and \yaref{thm:zorn}
+    Over $\ZF$ the \yaref{ax:c} and \yaref{thm:zorn}
     are equivalent.
 \end{remark}
 
@@ -84,7 +84,7 @@
 
 \begin{remark}[Cultural enrichment]
     Other assertion which are equivalent
-    to the axiom of choice:
+    to the \yaref{ax:c}:
     \begin{itemize}
         \item Every infinite family of non-empty sets
             $\langle  a_i : i \in I \rangle$
diff --git a/inputs/lecture_07.tex b/inputs/lecture_07.tex
index cf10aad..d1bbbe8 100644
--- a/inputs/lecture_07.tex
+++ b/inputs/lecture_07.tex
@@ -1,14 +1,5 @@
 \lecture{07}{2023-11-09}{}
 
-\begin{lemma}+
-    By the axiom of foundation there cannot exist infinite
-    descending chains
-    $x_1 \ni x_2 \ni x_3 \ni \ldots$
-\end{lemma}
-\begin{proof}
-    \todo{TODO!}
-\end{proof}
-
 \begin{notation}
     From now on, we will write $\alpha, \beta, \ldots$
     for ordinals.
diff --git a/inputs/lecture_08.tex b/inputs/lecture_08.tex
index 76874fe..25e55c8 100644
--- a/inputs/lecture_08.tex
+++ b/inputs/lecture_08.tex
@@ -165,7 +165,7 @@ Furthermore $F\,''a \coloneqq  \{y : \exists  x \in  a .~(x,y) \in F\}$.
     Using \AxC
     we get a function for $\langle A_x : x \in M \rangle$,%
     \footnote{Actually we only need the axiom of dependent choice,
-        a weaker form of the axiom of choice.
+        a weaker form of the \yaref{ax:c}.
         We'll discuss this later.% TODO REF
     }
     i.e.~a function $f\colon M \to  M$