TITLE: How should logical quantifiers be understood if the domain of discourse is not specified?
QUESTION [5 upvotes]: Let $P$ be a predicate. The expression $\forall x:P(x)$ seems to have two different meanings, depending on the context:

If the domain of discourse $D$ is clear from context, then the statement "$\forall x:P(x)$" should actually be read as an abbreviation of "$\forall x\in D:P(x)$". For instance, in real analysis the statement $\forall x:x^2\ge0$ would be understood as an abbreviation of $\forall x\in\mathbb R:x^2\ge0$.
In set theory, it seems that $\forall x:P(x)$ means "for any set $x$, $P(x)$ is true". For instance, the axiom of extensionality is written as $\forall x:\forall y:\forall z:(z\in x\iff z\in y)\implies x=y$.

I don't fully understand the meaning of $\forall x:P(x)$ in the context of set theory. Naïvely, the statement $\forall x:P(x)$ seems to be an abbreviation of $\forall x\in \mathbf{U}:P(x)$, where $\mathbf{U}$ is the universal set. However, the "universal set" is not actually a meaningful concept in standard formulations of set theory, and so this interpretation is clearly wrong. So how should the statement $\forall x:P(x)$ actually be interpreted?

REPLY [3 votes]: Actually, there is nothing wrong with understanding "$∀x ( Q(x) )$" in a foundational set theory to mean "$∀x{∈}U\ ( \ Q(x) \ )$" where $U$ is the type of all objects (i.e. $U$ denotes the entire intended domain). $U$ does not have to be a set. In pure ZFC, $U$ is not an object, because otherwise we get a contradiction. $U$ is commonly called a class. In extensions of ZFC that can reason about classes as objects, such as NBG or MK, we have two sorts, one for "sets" and one for "classes", and $U$ is still not a set, even though it is now an object.
Typically "$∀x{\in}S\ ( \ Q(x) \ )$" can be treated as a short-form for "$∀x\ ( \ x∈S ⇒ Q(x) \ )$", and "$x∈U$" simply reduces to "true". Note that the "$\in$" here does not require any set theory in a strict sense, because $S$ can just be a sort (in many-sorted FOL). However, it is convenient to use the same symbol as for set membership because there is no danger of ambiguity. After all, in NBG and MK we literally have $E ∈ \{ t : Q(t) \} ⇔ Q(E)$ where "$\{ t : Q(t) \}$" is the class notation.
In fact, this usage of quantifying over classes is extremely common in higher set theory, such as "$∀k{∈}ORD$" where $ORD$ is the class of ordinals, which is (as explained above) perfectly fine even in pure ZFC.