small fixes
All checks were successful
Build latex and deploy / checkout (push) Successful in 1m13s

This commit is contained in:
Josia Pietsch 2023-11-13 20:28:21 +01:00
parent 64a348389b
commit 26c32ce1bb
Signed by: josia
GPG key ID: E70B571D66986A2D
4 changed files with 5 additions and 14 deletions

View file

@ -10,7 +10,7 @@ $\ZFC$ stands for
\begin{itemize} \begin{itemize}
\item \textsc{Zermelo}s axioms (1905), % crises around 19000 \item \textsc{Zermelo}s axioms (1905), % crises around 19000
\item \textsc{Fraenkel}'s axioms, \item \textsc{Fraenkel}'s axioms,
\item the axiom of choice. \item the \yaref{ax:c}.
\end{itemize} \end{itemize}
\begin{notation} \begin{notation}
We write $x \subseteq y$ as a shorthand We write $x \subseteq y$ as a shorthand

View file

@ -14,7 +14,7 @@
Note further that if $b_1 \neq b_2$, Note further that if $b_1 \neq b_2$,
then $\{(b_1, x) : x \in b_1\} $ then $\{(b_1, x) : x \in b_1\} $
and $\{(b_2, x) : x \in b_2\}$ are disjoint. 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 $A$, gives us a choice function $f$ on $A$,
i.e.~$\forall b \in \cP(a) \setminus \{\emptyset\} .~(f(b) \in b)$. i.e.~$\forall b \in \cP(a) \setminus \{\emptyset\} .~(f(b) \in b)$.
@ -68,7 +68,7 @@
\end{refproof} \end{refproof}
\begin{remark} \begin{remark}
Over $\ZF$ the axiom of choice and \yaref{thm:zorn} Over $\ZF$ the \yaref{ax:c} and \yaref{thm:zorn}
are equivalent. are equivalent.
\end{remark} \end{remark}
@ -84,7 +84,7 @@
\begin{remark}[Cultural enrichment] \begin{remark}[Cultural enrichment]
Other assertion which are equivalent Other assertion which are equivalent
to the axiom of choice: to the \yaref{ax:c}:
\begin{itemize} \begin{itemize}
\item Every infinite family of non-empty sets \item Every infinite family of non-empty sets
$\langle a_i : i \in I \rangle$ $\langle a_i : i \in I \rangle$

View file

@ -1,14 +1,5 @@
\lecture{07}{2023-11-09}{} \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} \begin{notation}
From now on, we will write $\alpha, \beta, \ldots$ From now on, we will write $\alpha, \beta, \ldots$
for ordinals. for ordinals.

View file

@ -165,7 +165,7 @@ Furthermore $F\,''a \coloneqq \{y : \exists x \in a .~(x,y) \in F\}$.
Using \AxC Using \AxC
we get a function for $\langle A_x : x \in M \rangle$,% we get a function for $\langle A_x : x \in M \rangle$,%
\footnote{Actually we only need the axiom of dependent choice, \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 We'll discuss this later.% TODO REF
} }
i.e.~a function $f\colon M \to M$ i.e.~a function $f\colon M \to M$