Josia Pietsch
ca24c68790
Some checks failed
Build latex and deploy / checkout (push) Failing after 11m59s
34 lines
783 B
TeX
34 lines
783 B
TeX
\tutorial{}{2024-01-17}{}
|
|
|
|
\subsection{Sheet 9}
|
|
|
|
\nr 1
|
|
|
|
Let $\kappa$ be strongly inaccessible.
|
|
Then $(V_{\kappa}, \in \defon_{V_\kappa}) \models\ZFC$:
|
|
|
|
Most axioms are trivial.
|
|
|
|
\begin{itemize}
|
|
\item \AxUnion: Let $A \in V_{\kappa}$.
|
|
Then $\rank(x) < \kappa$ for all $x \in A$.
|
|
Since $\kappa$ is regular, we get
|
|
$\bigcup A \in V_{\kappa}$.
|
|
\item \AxPower: This holds since $\kappa$ is strongly inaccessible.
|
|
\item \AxRep: If $A \in V_{\kappa}$ and $f\colon A \to V_\kappa$
|
|
is definable over $V_\kappa$,
|
|
then $f'' A = \{f(a) : a \in A\}$ has bounded rank below $\kappa$.
|
|
\end{itemize}
|
|
|
|
|
|
|
|
\subsection{Exercise during tutorial}
|
|
|
|
|
|
Let $\kappa$ be uncountable and regular
|
|
Then the club filter $\cF_{\kappa}$ is $< \kappa$-closed.
|
|
|
|
|
|
|
|
|
|
|