# What is the class $\mathcal{W}$ in Moschovakis's "Notes on Set Theory"?

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?

• The author assumes that the theory of sets is "about something" that means that there is an initially unspecified domain of discourse with some sets and he call it that way. This amounts to say that the domain of discourse of the theory is non empty and with Axiom II we have that it contains at least one set: the empty one. Commented Nov 14, 2022 at 6:48
• The description of $\mathcal{W}$ says more than that. It says $\mathcal{W}$ is a class of sets which is “the standard universe of objects” and contains every model/universe of ZFC within it as a subclass (“natural universes are subuniverses of $\mathcal{W}$”)
– NikS
Commented Nov 14, 2022 at 9:15
• But initally the only assumption is about the non-emptiness of the universe... Later, after the axioms and the relevant theorems following them, we know much more than that: we know that the universe has certain specific features, and thus we can start with the meta-mathematical study of models of set theories. Commented Nov 14, 2022 at 9:41
• Sure, but saying that $\mathcal{W}$ is a collection of sets that satisfies some of the axioms still doesn’t describe $\mathcal{W}$ with particular specificity. For example, there are lots of models of ZFC, quite different from one another ($L$, forcing extensions of $L$, countable models of ZFC, etc.)
– NikS
Commented Nov 15, 2022 at 0:08

After some some additional reading, I've arrived at a tentative answer to my own question, namely that this notion of $$\mathcal{W}$$ is most likely intentionally non-specific (it's just some collection of sets satisfying axioms I-VI).