\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}