\lecture{04}{}{ZFC}

% Model-theoretic concepts and ultraproducts

\section{$\ZFC$}

% 1900, Russel's paradox
Russel's Paradox:
$\ZFC$ stands for
\begin{itemize}
    \item \textsc{Zermelo}’s axioms (1905), % crises around 19000
    \item \vocab{Fraenkel}'s axioms,
    \item the axiom of choice.
\end{itemize}
\begin{notation}
    We write $x \subseteq y$ as a shorthand
    for $\forall z.~(z \in x \implies z \in y)$.

    We write $x = \emptyset$ for $\lnot \exists  y . y \in x$
    and $x \cap y = \emptyset$ for $\lnot \exists  z . ~(z \in x \land z \in y)$.

    We use $x = \{y,z\}$
    for
    \[
    y \in x \land z \in x \land \forall a .~(a \in x \implies a = y \lor a = z).
    \]

    Let $x = \bigcup y$ denote
    \[
    \forall z.~(z \in x \iff \exists v.(v \in y \land z \in v)).
    \] 
\end{notation}
$\ZFC$ consists of the following axioms:
\begin{axiom}[\vocab{Extensionality}]
    \[
    \forall x.~\forall y.~(x = y \iff \forall z.~(z \in x \iff z \in y)).
    \]
    Equivalent statements using $\subseteq$:
    \[
        \forall x.~\forall y.~(x = y \iff (x \subseteq y \land y \subseteq x)).
    \]
\end{axiom}

\begin{axiom}[\vocab{Foundation}]
    Every set has an $\in$-minimal member:
    \[
    \forall  x .~ \left(\exists  a .~(a\in x) \implies
    \exists  y .~ y \in x \land \lnot \exists z.~(z \in  y \land z \in x)\right).
    \]
    Shorter:
    \[
    \forall x.~(x \neq \emptyset \implies \exists  y \in x .~ x \cap y = \emptyset).
    \]
\end{axiom}
\begin{axiom}[\vocab{Pairing}]
    \[
    \forall x .~\forall y.~ \exists z.~(z = \{x,y\}).
    \]
\end{axiom}
\begin{remark}
    Together with the axiom of pairing,
    the axiom of foundation implies
    that there can not be a set $x$ such that
    $x \in x$:
    Suppose that $x \in x$.
    Then $x$ is the only element of $\{x\}$,
    but $x \cap \{x\} \neq \emptyset$.

    A similar argument shows that chains like
    $x_0 \in x_1 \in  x_2 \in x_0$
    are ruled out as well.
\end{remark}

\begin{axiom}[\vocab{Union}]
    \[
    \forall x.~\exists y.~(y = \bigcup x).
    \]
\end{axiom}

\begin{axiom}[\vocab{Powerset}]
    We write $x = \cP(y)$
    for
    $\forall  z.~(z \in x \iff x \subseteq z)$.
    The powerset axiom (PWA) states
    \[
    \forall x.~\exists y.~y=\cP(x).
    \]
\end{axiom}
\begin{axiom}[\vocab{Infinity}]
    A set $x$ is called \vocab{inductive},
    iff $\emptyset \in x \land \forall y.~(y \in x \implies y \cup \{y\} \in x)$.

    The axiom of infinity says that there exists and inductive set.
\end{axiom}

\begin{axiomschema}[\vocab{Separation}]
    % TODO :(Aus)
    Let $\phi$ be some fixed
    fist order formula in $\cL_\in$.
    Then $\text{(Aus)}_{\phi}$
    states
    \[
    \forall v_1 .~\forall v_p .~\forall a .~\exists  b .~\forall x.~
    (x \in b \implies x \in a \land \phi(x,v_1,v_p))
    \]

    Let us write $b = \{x \in a |  \phi(x)\}$
    for $\forall x.~(x \in b \iff x \in a \land f(x))$.
    Then (Aus) can be formulated as
    \[
    \forall a.~\exists b.~(b = \{x \in a; \phi(x)\}).
    \]
\end{axiomschema}

\begin{notation}
    \todo{$\cap, \setminus, \bigcap$}
    % We write $z = x \cap y$ for $\forall u.~((u \in z) \implies u \in x \land u \in y)$,
    % $Z = x \setminus y$ for ...
    % $x = \bigcap y$ for ...
\end{notation}
\begin{remark}
    (Aus) proves that
    \begin{itemize}
        \item $\forall a.~\forall b.~\exists c.~(c = a \cap b)$,
        \item $\forall a.~\forall b.~\exists c.~(c = a \setminus b)$,
        \item $\forall a.~\exists b.~(b = \bigcap a)$.
    \end{itemize}
\end{remark}
\begin{axiomschema}[\vocab{Replacement} (Fraenkel)]
    Let $\phi$ be some $\cL_{\in }$ formula.
    Then
    \[
    \forall v_1 .~\exists  b.~\forall y.~(y \in b \iff \exists  x .~(x \in a \land \phi(x,y,v_1,v_p))).
    \]
\end{axiomschema}

\begin{axiom}[\vocab{Choice}]
    Every family of non-empty sets has a \vocab{choice set}:
    \begin{IEEEeqnarray*}{rCl}
        \forall x .~&(&\\
                    && ((\forall y \in x.~y \neq \emptyset) \land (\forall y \in x .~\forall y' \in x .~(y \neq  y' \implies y \cap y' = \emptyset))\\
                    && \implies\exists  z.~\forall y \in x.~\exists u.~(z \cap y = \{u\})\\
                    &)&
    \end{IEEEeqnarray*}
\end{axiom}


% TODO Hier weiter