On Page 64, Set Theory, Jech(2006), define the following, by transfinite induction:
- $V_0=\emptyset$,
- $V_{\alpha+1}=P(V_{\alpha})$,
- $V_{\alpha}=\bigcup_{\beta<\alpha}V_\beta$, if $\alpha$ is a limit ordinal.
How can we prove $V_{\alpha}$ is transitive by induction.
I tried what if other set, say $\{\{\emptyset\}\}$, is disignated as $V_0$. It turns out the transitive property fails. So somehow I should incoorperate $V_0=\emptyset$ into the induction. But then I stared at it, I stared at it, I just don't know what should I do.