Omega, well ordering and the cumulative hierarchy $V$.

Question

Hi I need help with a problem of set theory. I'm not sure how to prove that the well ordering on $\omega$ isomorphic to $\omega+\omega$ belongs to the level $V_{\omega+\omega}$ in the hierarchy $V$.

any help?

thanks!

Answer 1

Note that $\omega$ is a very particular set, it belongs $V_{\omega+1}$. It is not hard to calculate and see that $\omega\times\omega$ belongs to $V_{\omega+5}$ (or even less), so every subset of it would be in $V_{\omega+6}$ and in particular the subsets which are well-orders of any order type.

Answer 2

Any relation on $\omega$ is a subset of $\mathcal P (\mathcal P ( \mathcal P(\omega)))$.

$V_{\omega + \omega} = \bigcup_{\beta < \omega + \omega} \mathcal P (V_\beta)$ and $\omega \in V_{\omega + 1}$.

Now combine these two to obtain the desired.