41 lines
1.1 KiB
TeX
41 lines
1.1 KiB
TeX
\lecture{10}{}{} % Mirko
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Applications of induction and recursion:
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\begin{fact}
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For every set $x$ there is a transitive set $t$
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such that $x \in t$.
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\end{fact}
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\begin{proof}
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Take $R = \in $.
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We want a function $F$ with domain $\omega$
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such that $F(0) = \{x\}$
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and $F(n+1) = \bigcup F(n)$.
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Once we have such a function,
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$\{x\} \cup \bigcup \ran(F)$
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is a set as desired.
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\todo{insert formal application of recursion theorem}
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\end{proof}
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\begin{notation}
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Let $\OR$ denote the class of all ordinals
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and $V$ the class of all sets.
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\end{notation}
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\begin{lemma}
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There is a function $F\colon \OR \to V$
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such that $F(\alpha) = \bigcup \{\cP(F(\beta)): \beta < \alpha\}$.
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\end{lemma}
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\begin{proof}
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\todo{TODO}
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\end{proof}
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\begin{notation}
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Usually, one write $V_\alpha$ for $F(\alpha)$.
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They are called the \vocab{rank initial segments} of $V$.
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\end{notation}
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\begin{lemma}
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If $x$ is any set, then there is some $\alpha \in \OR$
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such that $x \in V_\alpha$,
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i.e.~$V = \bigcup \{V_{\alpha} : \alpha \in \OR\}$.
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\end{lemma}
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