gist complete

inputs/lecture_01.tex

@@ -109,6 +109,7 @@
     If $A \sim B$, there is a bijection $h\colon A \to B$.
 \end{theorem}
 \begin{proof}
+    \gist{%
     Let $f\colon A \hookrightarrow B$ and $g\colon B \hookrightarrow A$
     be injective.
     We need to define a bijection $h\colon A \to  B$.
@@ -146,6 +147,7 @@
     It is clear that this is bijective.
 
     \todo{missing picture $f(A^{\text{odd}}) \subseteq B^{\text{even}}$, $f(A^\infty) = B^\infty$}.
+}{Preimage sequence}
 \end{proof}
 
 \begin{definition}

inputs/lecture_03.tex

@@ -111,7 +111,7 @@ all condensation points are accumulation points.
     \] 
 \end{refproof}
 
-\todo{Alternative proof of Cantor-Bendixson}
+\gist{\todo{Alternative proof of Cantor-Bendixson}}{}
 % \begin{remark}
 %     There is an alternative proof of Cantor-Bendixson, going as follows:
 %     Fix $A \subseteq  \R$ closed.

inputs/lecture_04.tex

@@ -5,7 +5,7 @@
 \section{\texorpdfstring{$\ZFC$}{ZFC}}
 
 % 1900, Russel's paradox
-\todo{Russel's Paradox}
+% \todo{Russel's Paradox}
 $\ZFC$ stands for
 \begin{itemize}
     \item \textsc{Zermelo}’s axioms (1905), % crises around 19000

inputs/lecture_08.tex

@@ -108,10 +108,10 @@ Furthermore $F\,''a \coloneqq  \{y : \exists  x \in  a .~(x,y) \in F\}$.
 \end{axiom}
 
 
-\todo{notation: $\emptyset, \cap$}
+\gist{\todo{notation: $\emptyset, \cap$}}{}
 
 
-\todo{the following was actually done in lecture 9}
+\gist{(The following was actually done in lecture 9, but has been moved here for clarity.)}{}
 
 $\BGC$ (in German often NBG\footnote{\vocab{Neumann-Bernays-Gödel}})
 is defined to be  $\BG$

inputs/lecture_15.tex

@@ -60,7 +60,7 @@
 Recall the following:
 
 \begin{definition}
-    A substructure $X \subseteq  V_\theta$\todo{make this more general. Explain why $V_\theta$ is a model}
+    A substructure $X \subseteq  V_\theta$\gist{\todo{make this more general. Explain why $V_\theta$ is a model}}{}
     is an \vocab{elementary substructure}
     of $V_\theta$,
     denoted $X \prec V_{\theta}$,\footnote{more formally $(X,\in ) \prec (V_{\theta})$}

inputs/lecture_18.tex

@@ -77,6 +77,7 @@
     \end{enumerate}
 \end{theorem}
 \begin{proof}
+\gist{%
     2. $\implies$ 1.:
     Fix $j\colon  V \to M$.
     Let $U = \{X \subseteq  \kappa : \kappa \in j(X)\}$.
@@ -221,11 +222,60 @@ such that
 as otherwise $\forall \delta < \alpha. ~  \kappa \setminus X_\delta \in U$,
 i.e.~$\emptyset = (\bigcap_{\delta < \alpha} \kappa \setminus X_{\delta}) \cap X \in U \lightning$.
 We get $[f] = [c_{\delta}]$,
-so $\beta = \sigma([f]) = \delta([c_{\delta}]) = j(\delta) = \delta$,
+so $\beta = \sigma([f]) = \sigma([c_{\delta}]) = j(\delta) = \delta$,
 where for the last equality we have applied the induction hypothesis.
 So $j(\alpha) \le \alpha$.
 
+For all $\eta < \kappa$, we have $\eta = \sigma([c_{\eta}]) < \sigma([c_{\id}]) < \sigma([c_{\kappa}])$,
+so $j(\kappa) > \kappa$.
+
 % It is also easy to show $j(\kappa) > \kappa$.
+}{%
+    $2 \implies 1$:
+    Fix $j\colon  V \to M$. $U \coloneqq  \{X \subseteq  \kappa : \kappa \in j(X)\}$ is an UF:
+    \begin{itemize}
+        \item $M \models j(X \cap Y) = j(X) \cap j(Y)$,
+            hence $\kappa \in j(X) \cap j(Y) \implies \kappa \in j(X \cap Y)$
+        \item $M \ni X \subseteq Y \implies  Y \in M$, $\emptyset \not\in M$ (same argument)
+        \item $\kappa \in U$:
+            \begin{itemize}
+                \item $\forall \alpha \in \OR.~ j(\alpha) \in \Ord, j(\alpha) \ge \alpha$:
+                    \begin{itemize}
+                        \item Write $\alpha\in \OR$ and use $\alpha \in \OR \iff M \models j(\alpha) \in \OR$.
+                        \item Suppose $j(\alpha) < \alpha$, $\alpha$ minimal,
+                            but $M \models j(j(\alpha)) < j(\alpha) \implies j(j(\alpha)) < j(\alpha)$.
+                    \end{itemize}
+                \item $j(\kappa) \neq \kappa \implies j(\kappa) > \kappa \implies \kappa \in j(\kappa)$.
+            \end{itemize}
+        \item Ultrafilter: $\kappa \in j(\kappa) = j(X \cup (\kappa \setminus X)) = j(X) \cup j(\kappa \setminus X)$.
+        \item $< \kappa$ closed: For $\theta < \kappa$, $X_i \in U$:
+            $\kappa \in \bigcap_{i < \theta} j(X_i) = j\left( \bigcap_{i < \theta} X_i \right) \in U$.
+            ($j(\theta) = \theta$, so $j\left( \langle X_i : i < \theta \rangle \right) = \langle j(X_i) : i < \theta \rangle$.
+        \item Not principal:
+            $\xi < \kappa \implies j(\{\xi\}) = \{\xi\} \not\ni \kappa$.
+    \end{itemize}
+    $1 \implies 2$:
+    \begin{itemize}
+        \item Fix $U$. Consider $\leftindex^\kappa V$.
+        \item $f \sim g :\iff \{\xi < \kappa: f(\xi) = g(\xi)\} \in U$.
+        \item $[f] \coloneqq \{g : g \sim f \land g \in V_\alpha \text{ for minimal } \alpha\}$ (Scott's Trick).
+        \item $[f] \tilde{\in } [g] :\iff \{\xi < \kappa: f(\xi ) \in g(\xi)\}  \in U$.
+        \item \yaref{thm:los}: $(\cF, \tilde{\in }) \models \phi([f_1], \ldots, [f_k]) \iff \{\xi < \kappa: (V, \in ) \models \phi(f_1(\xi), \ldots, f_k(\xi))\} \in U$.
+        \item $\overline{j}(x) \coloneqq  [ \xi \mapsto x]$.
+        \item $\tilde{\in }$ well-founded: lift decreasing sequences ($U$ is $<\kappa$ closed, $ \omega < \kappa$)
+        \item $\tilde{\in }$ set-like $\overset{\yaref{thm:mostowksi}}{\leadsto}$ $(\cF, \tilde{\in }) \overset{\sigma}{\cong} (M, \in )$.
+        \item $j \coloneqq  \sigma \circ \overline{j}$.
+        \item $\alpha < \kappa \implies j(\alpha) = \alpha$ (know $j(\alpha) \ge \alpha$):
+            \begin{itemize}
+                \item Induction for $j(\alpha) \le \alpha$: Fix $\alpha$, $\sigma([f]) = \beta \in j(\alpha)$.
+                \item $[f] \tilde{\in }[c_\alpha]$.
+                \item $\exists  \delta < \alpha.~[f] = [c_\delta]$ ($U $ is $<\kappa$ closed)
+                \item $\beta = \sigma([f]) = \sigma([c_\delta]) = j(\delta) \overset{\text{IH}}{=} \delta \in \alpha$.
+            \end{itemize}
+        \item $j(\kappa) \neq \kappa$:
+            $\forall \eta < \kappa.~\eta = \sigma([c_{\eta}]) < \sigma([\id]) < \sigma([\kappa])$.
+    \end{itemize}
+}
 \end{proof}
 
 \begin{theorem}[\L o\'s]

inputs/lecture_22.tex

@@ -137,7 +137,7 @@ is well founded.
             so $M[g] \models \text{``$\{x,y\}$ is the pair of $x$ and $y$''}$.
             Hence $M[g] \models \AxPair$.
         \item \AxUnion:
-            Similar to \AxPair.\todo{Exercise}
+            Similar to \AxPair.\gist{\todo{Exercise}}{}
     \end{itemize}
 \end{proof}