The cumulative hierarchy and the real numbers

[hmmmmm]

We define the cumulative hierarchy as:

$V_0=\emptyset$

$V_{\alpha+1}=\mathcal{P}(V_\alpha)$

If $\lambda$ is a limit ordinal then $V_\lambda=\bigcup_{\alpha<\lambda} V_\alpha$

Then we have a picture of a big V where we keep building sets up from previous ones and each $V_\alpha$ is the class (set?) of all sets formed from the previous stages.

Now I am wondering how we get from here to a construction of the real numbers? I can see that we will have a set of size $|\mathbb{R}|$ by $V_{\omega+2}$ and then we could go on to construct the reals formally via dedekind cuts of cauchy sequences. However are the sets in the hierarchy well founded in which case $\mathbb{R}$ would not be there?

Thanks for any help

 

[hmmmmm]

I'm not too sure how to mark a thread as solved or something but my confusion here came from thinking that unions and power sets preserved well ordering, which they do not