Two cumulative hierarchies

Question

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?

Answer 1

There are many many many different ways to describe models of $\sf ZFC$ via hierarchies. In particular in "constructible" models, a-la $L$, you have the following (of course an arbitrary model of $\sf ZFC$ may not have the second and third hierarchies):

1. The cumulative (von Neumann) hierarchy, $V_\alpha$.
2. The constructible hierarchy, $L_\alpha$.
3. The fine structure hierarchy, $J_\alpha$.
4. The hierarchy without a name $H(\omega_\alpha)$.

Different hierarchies are used for different objectives. We use the constructible hierarchy to define a well-order of the universe, but we use the cumulative hierarchy to define the rank of $\in$, which we later use for epsilon induction and recursion.

Your assessment is incorrect, though. Note that $\omega+2\notin V_{\omega+1}$, despite being hereditarily countable. Many other counterexamples can be easily produced using similar points.

Often, by the way, we don't really care about the finite sets, so there is no issue in lumping them all together into one first-level basis and continuing from there.

If we return to the regular definitions, then one can show the following:

1. If $\delta$ is a limit ordinal then $V_\delta$ is a model of the axioms of $\sf ZFC$, with the possible exception of replacement axioms. In fact it is a model of Zermelo's set theory (and slightly more), which is sufficient to develop a lot of modern mathematics.
2. If $\kappa$ is a regular cardinal then $H(\kappa)$ satisfies all the axioms of $\sf ZFC$ except power set, unless $\kappa$ is also a strong limit cardinal.

So now suppose that you want a model of a subtheory of $\sf ZFC$ which satisfies "enough" axioms, you can choose some $\kappa$ large enough and then use $V_\kappa$ or $H(\kappa)$ for your convenience (and often, countable elementary submodels).

Note that if $\varphi(x,\alpha)$ is a formula such that $\alpha$ must be an ordinal, and:

• $T_\alpha=\{x\mid\varphi(x,\alpha)\}$ is a set for every $\alpha$.
• $\forall x\exists\alpha\varphi(x,\alpha)$.

Then $\varphi$ defines a hierarchy, $T_\alpha$. We say that the hierarchy is continuous if for limit ordinal $\beta$ it holds $T_\beta=\bigcup_{\alpha<\beta}T_\alpha$.

Whether or not the property described by $\varphi$ is useful to us is an open-ended question, certainly there will be useless hierarchies and there be useful ones, but we can't really know which ones are the useless because their uses may not be apparent to us.