The Hole Argument Against Worlds with Domains

More or less every view about the nature of possible worlds that I'm familiar with develops a theory which assigns to each world $w$ a domain $D_w$ of objects which "exist" at $w$. I want to argue that every such view faces a Hole Argument analogous to the one that appears in debates about space and time.

Let ${\mathsf{P}}_1$ and ${\mathsf{P}}_2$ be two properties. For our purposes, it doesn't matter what they are. There might be 97 properties, along with 34 relations. But that would complicate the discussion needlessly. For definiteness and vividness, let the properties be $\mathsf{Red}$ and $\mathsf{Green}$. (Again, you might want to consider some fancy quantum properties, but nothing hinges on this.)

Let us suppose, with standard metaphysics of worlds, that we consider a world $w$ with a domain,

$D_w=\{a,b,c\}$

such that,

$\mathsf{Red}(a)\wedge \mathrm{\neg }\mathsf{Red}(b)\wedge \mathrm{\neg }\mathsf{Red}(c)$

$\mathsf{Green}(a)\wedge \mathsf{Green}(b)\wedge \mathrm{\neg }\mathsf{Green}(c)$

It follows that the extensions of $\mathsf{Red}$ and $\mathsf{Green}$ are:

${\mathsf{Red}}^w=\{a\}$
${\mathsf{Green}}^w=\{a,b\}$

Let us now permute the domain under the bijection

$\pi :D_w\to D_w$

given by:

$\pi (a)=b$
$\pi (b)=c$
$\pi (c)=a$

And let us apply this permutation $\pi$ to the extensions above, obtaining:

$\pi [{\mathsf{Red}}^w]=\{b\}$
$\pi [{\mathsf{Green}}^w]=\{b,c\}$

Now, we still have the same domain $D_w$, but we now have distinct extensions.

Question: is the result a new distinct world? Or ...?

On the standard view, the result is a new distinct world $w^{\prime }=w^{\pi }$ with the same domain $D_w$ but such that,

${\mathsf{Red}}^{w^{\prime }}=\{b\}$
${\mathsf{Green}}^{w^{\prime }}=\{b,c\}$

But the problem is that this violates anti-haecceitism. For $w$ and $w^{\prime }$ are isomorphic (under $\pi$, by construction), and yet extensionally distinct. Anti-haecceitism (along with Leibniz Equivalence in space-time physics) states that somehow $w$ and $w^{\prime }$ should be the same world.

Suppose one accepts anti-haeccetism. Then, there seem to me to be only two ways out of this problem:

(i) Keep the domain-based notion of worlds, and deny that the world $w^{\prime }$ exists.
(ii) Throw away the domain, and insist that we have two descriptions of the same world.

The first seems to me to be more or less Lewis's view. But I think the second view is much more attractive, and it fits very naturally with physics. On the view I prefer, this world $w$ is one at which there are exactly 3 concreta $x_1,x_2,x_3$ and such that

$\mathsf{Red}(x_1)\wedge \mathrm{\neg }\mathsf{Red}(x_2)\wedge \mathrm{\neg }\mathsf{Red}(x_3)$

$\mathsf{Green}(x_1)\wedge \mathsf{Green}(x_2)\wedge \mathrm{\neg }\mathsf{Green}(x_3)$

But there is no special domain for $w$. And the attempt to label the "objects" is really a form of skolemization. One can skolemize the description of the world, if one wants; and one can take the constants as the domain elements. (Alternatively one can consider a model $\mathcal{A}$ of the world $w$.) But this is no longer the world $w$, but a representation of the world $w$.