What does Skolem's paradox have to do with "unrelativized quantification"?

Question

Timothy Bays' excellent essay on Skolem's paradox includes the following claim:

... Skolem's Paradox doesn't introduce contradictions into various forms of axiomatized set theory, even when these axiomatizations are themselves understood formalistically or model-theoretically. From a proof-theoretic standpoint, for example, there is a difference between unrelativized quantification and quantification which has been explicitly relativized to some formula in our language (where this formula is one that, from an intuitive perspective, serves to “pick out” the domain of countable model of ZFC). So, there's no a priori reason to think that a sentence with unrelativized quantifiers will conflict with that sentence's fully-relativised counterpart.

What does he mean by "relativized" or "unrelativized quantification" from a "proof-theoretic standpoint"? Later, in the same paragraph, he speaks about domain-restricted quantification, so it cannot be that.

Answer 1

Suppose that $V$ is our universe, and $A$ is a subset of that universe. We can relativise the formulas to $A$ by recursion:

1. If $\varphi(\vec x)$ is an atomic formula, then $\varphi^A(\vec x)$ is simply $\varphi(\vec x)\land\vec x\in A$. (Here $\vec x\in A$ is understood as the conjunction of $x_i\in A$ for the variables appearing in $\vec x$).

2. If $\varphi=\varphi_0*\varphi_1$ (or $\lnot\varphi_0$), then $\varphi^A$ is $\varphi_0^A*\varphi_1^A$ (or $\lnot\varphi_0^A$), where $*$ is any connective.

3. If $\varphi=\exists x\varphi_0(x)$, then $\varphi^A=\exists x(x\in A\land\varphi_0(x))$.

Of course, if $A$ is not definable, we need to add it as a predicate. We can deduce from this that $(\forall x\varphi(x))^A=\forall x(x\in A\to\varphi^A(x))$, assuming the logic we're working with is close enough to first-order logic.

The point is that since ZFC proves that $\Bbb R$ is uncountable, it proves "for all $x$, if $x$ is a function from $\Bbb N$ to $\Bbb R$, then $x$ is not surjective". When we take a countable model of ZFC, the quantifiers (and the interpretation of $\Bbb N$ and $\Bbb R$) all relativise to that model. That means that we simply have:

For all $x$ in the model, if $x$ is a function from $\Bbb N$ of the model to $\Bbb R$ of the model, then $x$ is not surjective (in the model).

Note that we can have a reverse Skolem paradox: if ZFC is consistent, then it has a model $M$ such that $\Bbb N^M$ is uncountable (externally!).