This commit is contained in:
parent
64a348389b
commit
26c32ce1bb
4 changed files with 5 additions and 14 deletions
|
@ -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
|
||||||
|
|
|
@ -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
|
||||||
$\langle a_i : i \in I \rangle$
|
$\langle a_i : i \in I \rangle$
|
||||||
|
|
|
@ -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.
|
||||||
|
|
|
@ -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 $f\colon M \to M$
|
||||||
|
|
Loading…
Reference in a new issue