w23-logic-2/inputs/lecture_10.tex

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2023-11-23 15:38:28 +01:00
\lecture{10}{}{} % Mirko
Applications of induction and recursion:
\begin{fact}
For every set $x$ there is a transitive set $t$
such that $x \in t$.
\end{fact}
\begin{proof}
Take $R = \in $.
We want a function $F$ with domain $\omega$
such that $F(0) = \{x\}$
and $F(n+1) = \bigcup F(n)$.
Once we have such a function,
$\{x\} \cup \bigcup \ran(F)$
is a set as desired.
\todo{insert formal application of recursion theorem}
\end{proof}
\begin{notation}
Let $\OR$ denote the class of all ordinals
and $V$ the class of all sets.
\end{notation}
\begin{lemma}
There is a function $F\colon \OR \to V$
such that $F(\alpha) = \bigcup \{\cP(F(\beta)): \beta < \alpha\}$.
\end{lemma}
\begin{proof}
\todo{TODO}
\end{proof}
\begin{notation}
Usually, one write $V_\alpha$ for $F(\alpha)$.
They are called the \vocab{rank initial segments} of $V$.
\end{notation}
\begin{lemma}
If $x$ is any set, then there is some $\alpha \in \OR$
such that $x \in V_\alpha$,
i.e.~$V = \bigcup \{V_{\alpha} : \alpha \in \OR\}$.
\end{lemma}