\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.