w23-logic-2/inputs/tutorial_2024-01-17.tex

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2024-01-17 11:51:59 +01:00
\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.