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

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!

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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.

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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.
Any pair $(n,m)$ is in $V_\omega$, so $\omega\times\omega$ and all its subsets are in $V_{\omega+1}$. –  Andres Caicedo Apr 22 '13 at 18:01
This is a test$\vphantom{@Arthur}$. Did you get pinged? :-) –  Asaf Karagila May 5 '13 at 22:28