Why does $\exists x\,\ x = x$? The Wikipedia article on ZFC insists that the empty set exists since it suffices for any set to exist, since the Axiom of Specification for which we always specify "false" will construct the empty set.
I agree that in the end everything works out because the Axiom of Infinity guarantees the existence of $\mathbb{N}$, but apparently this is not needed.
Directly paraphrasing the article, ZFC is "formalized," the "domain of discourse" must be nonempty, and therefore $\exists x\,\ x = x$. I don't completely understand this. Why is the assertion $\exists x$ even true (reflexivity I can accept)?
 A: The formula $x=x$ (where $x$ is a variable) is an axiom of first-order logic, in which set theory is formulated. By generalisation it follows that $\forall x\ x=x$ is true... so if a set exists at all then it must satisfy $x=x$. Since the axiom of infinity asserts that a set exists, it must follow that $\exists x\ x=x$ is a true statement.
As for the empty set, its existence follows from the axioms of infinity and separation. The axiom of infinity gives you that a set exists... let's call it $\omega$. The axiom of separation tells you that the following is true
$$\exists x[y \in x\ \leftrightarrow\ y \in \omega\ \wedge\ y \ne y]$$
but any witness $x$ to this sentence must be the empty set.
A: It is a convention that domains of discourse should always nonempty (or syntactically $\exists x: \top$ should be a tautology).
If you reject this convention, then ZFC with the axioms of infinity and empty set omitted does indeed have an empty model.
It is by no means necessary to adopt this convention; in fact, there are quite good reasons to reject it. In my experience, texts either adopt or reject this convention without comment, so it is easy enough to go for many years without even being aware that there are split opinions on this. To wit, my participation in the talk page for the very article you link was the first time I had ever heard of it.
It is wikipedia's convention, as far as I can tell, to reserve the phrase "first-order logic" to refer to first-order logic with the adoption this convention, and use the phrase "free logic" refer to first-order logic without this convention. (that talk page was also the first time I had ever heard the phrase "free logic")
