Note: Some further reading led me to sections of Moschovakis's "Notes on Set Theory" which may narrow this down to a more specific question, posted here
Just & Weese's "Discovering Modern Set Theory" Chapter 12 (relevant excerpt here) begins with: "Imagine you are the architect of the universe of sets. Your task is to build up this universe step by step…how will you proceed?" It then walks the reader through the well-known definition of $V_n$ for finite ordinals $n$, as well as the concept of $V_{\omega} = \bigcup_{n\in\omega}V_n$, and finally presents the following definition for the desired universe:
- $V_0 = \emptyset$
- $V_{\alpha+1} = \mathcal{P}(V_{\alpha})$
- $V_{\alpha} = \bigcup_{\beta<\alpha}V_{\beta}$ for $\alpha \in \mathbf{LIM}$
- The desired universe of sets is $V=\bigcup_{\alpha \in \mathbf{ON}}V_{\alpha}$
Other sources develop an essentially identical definition of "the universe of sets" $V$, for example:
- Keith Devlin's The Joy of Sets (relevant excerpt here)
- Stanford Encyclopedia of Philosophy article on set theory (written apparently by Joan Bagaria)
This leads to a sort of conceptual question of what such a von Neumann hierarchy-style definition of the "universe of sets" actually means. Two possibilities come to mind:
The first is that you are starting with nothing but the primitive notions ("set" and $\in$) and the axioms of ZFC, and then you start asserting the existence various sets that you've defined in conformity with the axioms (ordinals, the levels $V_{\alpha}$ of the hierarchy) until you have "built up" your universe $V$.
The second is that you aren't starting with nothing, but rather you are starting with an implicit (unstated but assumed) "preliminary universe" consisting of (presumably) "every possible set" (maybe even non-well-founded sets?). And then the definition of $V$ is thought of as a process of iteratively "picking out" and adding to $V$ various sets that already "exist" in the "preliminary universe". (FWIW, the development of "the set universe" $V$ in the books/website mentioned above either make no mention of such a "premliminary universe", or at most only gesture vaguely at such a concept)
I'm not sure if there's a big material difference between these two points of view. But to the extent that it matters:
- Which of these two concepts is the "preferred" one? Or maybe neither (if some third concept is preferable)?
- And if it's the second one: What exactly is the "preliminary universe", and can it be described more precisely than the somewhat vague phrase "every possible set"? Also, what do you call this "preliminary universe" so that people will know what you're talking about? ("preliminary universe" was obviously a term I totally made up)
Addendum:
In this article by Timothy Chow, section 3.2 refers to "$V$, the class of all sets". Later (section 4), in discussing set theory as the foundation of all mathematics, he says "the class $V$ of all sets may be thought of as being the entire mathematical universe". This certainly sounds like he's thinking of $V$ as a particular set universe, and not just any old set universe satisfying the axioms of ZFC.
This seems reasonable, since it seems implausible that every universe of ZFC can truly represent "the entire mathematical universe." For example (to take an extreme case) surely Cohen's minimal model of ZFC can't represent "the entire mathematical universe", given that it's a countable model. More generally, given two different set universes $V_A$ and $V_B$ ($V_A \neq V_B$), it's hard to see how they can both represent "the entire mathematical universe": by hypothesis, $V_A$ lacks some sets that are present in $V_B$ (we can always swap labels A & B to make this true), so if $V_B$ represents "the entire mathematical universe" then the same cannot be said of $V_A$ (it is missing some stuff).
So when Chow writes "the class $V$ of all sets may be thought of as being the entire mathematical universe", which particular class $V$ is he referring to?
Best Answer
The construction of the von Neumann hierarchy is used in two ways:
We want to show that adding the axiom of regularity will not add inconsistencies to the rest of the axiom. For that, if $W$ was a universe that satisfied all the axioms of $\sf ZF$ without regularity, we construct $V$ inside of it, and show that it is a model of $\sf ZF$. We can prove more, for example, inside $W$, this is the largest transitive class which is a model of $\sf ZF$.
In analysing a model of $\sf ZF$, the von Neumann hierarchy is an internal filtration of the universe. Namely, every set is captured by some stage; it has nice properties; and it is definable and internal. So the model "recognises" it.
While the first use is important, and indeed often presented first (for obvious reasons: we want to argue that axioms that seem "odd" do not introduce inconsistencies to the others), it is a one-time use for the von Neumann hierarchy and the vast majority of its uses are indeed of the second type.
But now we can talk about different models of set theory. Perhaps we have one satisfying Gödel's constructibility axiom ($V=L$, as it is often shorthanded), or maybe we have a different universe, satisfying some large cardinal axioms which contradict the constructibility axiom.
Each of those has its own von Neumann hierarchy. Each of these models, of these universes of sets, will "think" that they contain all the sets, and each of them will understand its own von Neumann hierarchy through this lens. And it may very well be that some of these universes will also have a different internal filtration, e.g. in the case of $L$ we have the $L_\alpha$ hierarchy; in other situations we can have a different filtration related to a concept called "fine structure" and "core models"; all of these can also be relativised in numerous ways.
But, again, each universe of set theory thinks of itself as being "the" universe of "all sets", and each one understands itself through its von Neumann hierarchy.
Now, it can very well be that two universes share some part of their von Neumann hierarchy. Or it could be that one universe is quite literally an initial segment of another (in that it is actually just $V_\alpha$ in the larger universe). These phenomena are studied in set theory through forcing, inner models, and other related concepts.