I was just reading the Wikipedia page on the Von Neumann universe, where it is stated that this universe "is often used to provide an interpretation or motivation of the axioms of ZFC." However, later on in the article, the Von Neumann universe is formally constructed using "sets," starting with "the empty set." This bothers me greatly, because as I understand it, ZFC is merely a theory describing how the "sets" in a model must behave; it is not itself a model.
For example ZFC, does not specify the particular object that the "empty set" is; it merely stipulates that its models must satisfy the axiom of the empty set (through either including this axiom directly or deducing it from the other axioms), i.e., the axiom,
$$ \exists x\forall y(\lnot y\in x) \text. $$
So, how is it possible to construct the Von Neumann universe from the very theory it's supposed to be a model for? Where does the definition of the empty set actually come in? I would appreciate any clarification.
Update: Asaf Karagila writes here that
If you want to consider the "simple" foundational approach to theories like $\sf ZFC$, then sets are primitive objects and $V$ is a given universe to begin with.
The axioms of $\sf ZFC$ tell you what sort of properties $V$ and its $\in$ relation satisfy. [...]
What the von Neumann hierarchy gives you is the understanding that if $V$ is already given, then we can write this wonderful filtration of $V$ into a very nice hierarchy. Additional theorems like the reflection theorem also tell you more about this hierarchy and its deep connection with the structure of $V$ as a universe of sets.
So am I correct in understanding that the Von Neumann universe is a construction that we put together once we assume that we have some model $V$ of ZFC?