This question relates to Yiannis Moschovakis's book Notes on Set Theory (first edition, ebook of which can be "borrowed" at archive.org). Relevant (limited) excerpts posted here
Section 3.6 starts as follows (he allows for the possibility of atoms, a.k.a. urelements, but for simplicity I ignore that case):
The axiomatic setup. We assume at the outset that there is a domain or universe $\mathcal{W}$ of objects, some of which are sets, and certain definite conditions and operations on $\mathcal{W}$, among them the basic conditions of identity, sethood, and membership:
$x = y \longleftrightarrow x$ is the same object as $y$
$Set(x) \longleftrightarrow x$ is a set
$x \in y \longleftrightarrow Set(y)$ and $x$ is a member of $y$
We call the objects which are not sets atoms, but we do not require that any atoms exist, i.e. it may be the case that all the objects are sets.
Section 8.20 goes on to say the following:
We have assumed at the outset, in 3.6, that our theory has a model, the standard universe of objects $\mathcal{W}$, in which axioms (I) - (VI) (at least) are true. This assumption is natural and even necessary if our lives as set theorists are to have any meaning...When we assert the existence of models of various theories, to construct those, we have to start with something, and that is always the assumed, standard model of our theory.
(note that the axioms I-VI in this book are maybe more commonly formulated as seven axioms: Empty Set, Extensionality, Comprehension, Pair, Union, Power Set, and Infinity)
Finally, Appendix B (Axioms and Universes) defines a "natural set universe" to be essentially (in a theory without urelements) a transitive class of sets, and then states:
When we view a set universe $\mathcal{M}$ as a model for an axiomatic set theory, we take its definite conditions to be all the definite conditions of our basic domain $\mathcal{W}$...From the mathematical point of view, natural universes are subuniverses of $\mathcal{W}$ and they inherit their structure from $\mathcal{W}$, much like subgroups, subposets, topological subspaces and the like are specified by a subset of some given space and inherit the relevant structure from it.
So, what Moschovakis seems to be describing here is a sort of "super-universe" of sets, i.e. a particular class of sets $\mathcal{W}$ such that any other model/universe of ZFC is a subclass of $\mathcal{W}$. Now, that all makes sense as far as it goes, but what I don't quite get is that this "super-universe" $\mathcal{W}$ is never really described with any specificity.
So that's the question: What exactly is $\mathcal{W}$? In the context of "pure" set theory (without urelements), you could, I suppose, just say that it's the class consisting of "every concievable" set. But that's kind of vague. Is there some way to more precisely describe this $\mathcal{W}$ that Moschovakis speaks of?
