Home > Mathematical Logic, Set Theory > The Cumulative Hierarchy of Sets

## The Cumulative Hierarchy of Sets

The way the von Neumann cumulative hierarchy of sets is usually presented is the following:

1. $V_0 = \emptyset$
2. (Sucessor case) $V_{\alpha + 1} = {\mathcal P}(V_\alpha)$
3. (Limit case) $V_\alpha = \bigcup{\{V_\beta : \beta < \alpha\}}$

Then, $\textbf{V} = \bigcup{\{V_\alpha : \alpha \in \textbf{ON}\}}$.

Let us examine this definition in slow motion.

Informally, we think of each $V_\alpha$ as a small universe of sets. We think of sets as containers, uniquely identified by their content (I took this idea of contents and containers from Boolos and Jeffrey, Computability and Logic).

We start from nothing. $V_0$ is a universe where no sets exist, not even the empty set. $\lnot \exists x(x=x)$ is true in $V_0$.

So we have already built one content: the empty one. And we are willing to attach a container to this content: the empty set $\emptyset$, but we cannot yet, that’s the task of the powerset operation.

The powerset operation applied on $V_\alpha$ supplies us with new content by attaching a container, i.e. a set, to each possible content made of what we have obtained so far, i. e. to each possible combination of sets living in $V_\alpha$. This collection of sets is, then, the content of the following universe $V_{\alpha + 1}$. Actually not all of this content is entirely new, but that’s why we call the hierarchy of the $V_\alpha$s “cumulative”.

So, the empty set will be the only content of our next universe $V_1$. Note that $V_1$ knows $V_0$ is a set, but $V_0$ doesn’t. This holds in general for any $V_{\alpha + 1}$ and $V_\alpha$.

The union operation supplies us with a new content by collecting every set previously obtained by iteration of the powerset operation. Note that no container is attached to any content with this operation (no new sets are formed), only an entirely new content is generated. The subsequent application of the powerset operation to this new content is what is going to do the job of attaching a set to it.

Hence, for example, $V_\omega$ is the set of hereditarily finite sets, but this set, like $\omega$, and their subsets, functions and relations, and in fact any other infinite set of hereditarily finite sets, lives in $V_{\omega + 1}$. In turn, ${\mathcal P}(\omega)$, like $V_{\omega + 1} = {\mathcal P}(V_\omega)$, lives in $V_{\omega + 2}$.

Note that this process goes on as long as there are ordinals available. Successors mark the places where we can apply the powerset operation to generate new sets and limits the places where we can look back and see what we have done.