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

The Cumulative Hierarchy of Sets

January 12, 2011 Leave a comment Go to comments

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, although I use it differently here).

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_\alphas “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, one not available previously, by collecting the content of 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}.

This process goes on as long as there are ordinals available. The definition takes them for granted. Successors mark the places where we can apply the powerset operation to generate new sets and limits the places where we can look back to see what has been done so far.

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