small changes

inputs/lecture_07.tex

@@ -9,7 +9,7 @@
     \begin{enumerate}[(a)]
         \item $0$ is an ordinal, and if $\alpha$ is
             an ordinal, so is $\alpha + 1$.
-        \item If $\alpha$ is an ordinal and $x \in a$,
+        \item If $\alpha$ is an ordinal and $x \in \alpha$,
             then $x$ is an ordinal.
         \item If $\alpha, \beta$ are ordinals
             and $\alpha \subseteq \beta$,

inputs/lecture_09.tex

@@ -54,7 +54,7 @@ An alternative way of formulating this is
     Let $D$ be a class of triples
     such that for all $u,x$ there is exactly
     one $y$ with $(u,x,y) \in D$
-    (basicalls $(u,x) \mapsto y$ is a function).
+    (basically $(u,x) \mapsto y$ is a function).
 
     Then there is a unique function $f$ on $A$
     such that for all $x \in A$,

inputs/lecture_10.tex

@@ -158,7 +158,7 @@ This $F$ is called the \vocab{rank function} for $(A, R)$.
     \]
     and
     \[
-    \rk_R \rank(R) \coloneqq \ran(F).
+    \rank(R) \coloneqq \ran(F).
     \]
 
     In the special case that $R$ is a linear order on $A$,