In Timothy Bays' excellent article on Skolem's paradox, I have been stumped by the following lines:
Let's grant that the specific element which serves as “the empty set” will not remain constant as we move from one model of set theory to another—with the empty set in the first model becoming, perhaps, a singleton in the second. Nonetheless, we can still use a formula in the language of set theory to capture the notion “x is the empty set” in an essentially absolute way. In any model of our axioms, an element m' ∈ M will satisfy the open formula “∀y y ∉ x” if and only if the set {m | M ⊨ m ∈ m' } is really empty. Hence, there's at least a sense in which we can still capture the notion “x is the empty set” from within the algebraic framework. And this point extends more widely—a similar argument would apply to notions like “x is a singleton” or “x has seventeen members.” Even on the algebraic conception of axiomatization, therefore, there are some set-theoretic notions which we can still pin down pretty precisely.
At first he says that the element for "the empty set" will not remain constant from one model to another, but then he goes on to show that the empty set is really the self-same empty set in all models. What am I missing here?
This is the difference between intensional and extensional definitions. We like to think of sets as classes, i.e. every set is definable (remember that we allow parameters, so $A$ is given by the class $x\in y$, where $A$ is taken as the parameter $y$).
But in reality that can easily fail when the model's $\in$ relationship is not the "real $\in$".
The internal empty set is simply the object $\varnothing^M$ (in $M$) such that $M\models\forall x(x\notin\varnothing^M)$. We can show that it is unique in $M$, using the axiom of extensionality. But we can also talk about the empty class, which is the subset of $M$ which corresponds to the elements of $\varnothing^M$ inside $M$. So the empty class of $M$ is always empty, since $M$ thinks that there are no elements inside $\varnothing^M$.
What we do, normally, is identify a set with its class, so $\{x\in M\mid M\models x\in\varnothing^M\}$ is identified with $\varnothing^M$, even if it's not equal to it.
However, since the empty set defines the empty class, and that is externally empty, it means that the empty class of all models is really just the empty set.
Finally, let me now point out that every model of $\sf ZF$ has an initial segment which is truly well-founded, so by Mostowski's collapse lemma we can collapse that initial segment. It is easy to show that this initial segment must always include $V_\omega$, so it includes all the truly hereditarily finite sets, which of course include $\varnothing$.
So in a way, $\varnothing$ is the same in all models, up to a small re-labeling issue.