I'm trying to understand the purpose of
- the rank of a set, and more generally
- the cumulative hierarchy.
And although the comments left there are good, I find myself wanting a deeper understanding.
Now in general, a good way of understanding why things are the way they are, is to ask: "Why aren't things some other way?" Hence the following question.
Assume the usual axioms of ZFC, but do not assume GCH. In fact, keep in mind the possibility that GCH is false. Furthermore, let $H(\kappa)$ denote the collection of all sets that are hereditarily of cardinality $<\kappa$. For the purposes of this answer, lets also redefine the aleph and beth numbers. In particular, lets define that $\aleph_0 = \beth_0 = $ the very least cardinal number, as opposed to merely the least infinite cardinal number. Namely $0$.
Now define two ordinal-indexed sequences $U_\alpha$ and $V_\alpha$. In particular:$$U_\alpha = H(\aleph_\alpha),\,V_\alpha=H(\beth_\alpha).$$
It seems obvious that the sequence $V_\alpha$, as defined above, is the usual cumulative hierarchy. (Can someone verify this??)
Question. Is there any reason why the sequence $U_\alpha$ would be more/less useful than $V_\alpha$? For instance, if we define the $U$-rank of a set $X$ to be the least $\alpha$ such that $X \in U_\alpha$, and the $V$-rank of a set $X$ to be the least $\alpha$ such that $X \in V_\alpha$, is there any reason why the $U$-rank of a set would be more/less useful than the $V$-rank?
More generally: what are the specific properties of the usual rank and/or the usual cumulative hierarchy that make it useful and interesting?