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