MHB The cumulative hierarchy and the real numbers

AI Thread Summary
The cumulative hierarchy is defined by starting with an empty set and recursively forming new sets through power sets and unions at limit ordinals. The discussion explores how this hierarchy leads to the construction of real numbers, particularly through Dedekind cuts or Cauchy sequences. There is uncertainty about whether the sets in the hierarchy are well-founded, which could affect the existence of real numbers within this framework. The confusion arises from the misconception that unions and power sets preserve well-ordering, which they do not. Clarification on these points is sought to better understand the relationship between the cumulative hierarchy and the real numbers.
hmmmmm
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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
 
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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
 
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