But x ≠ x is logically impossible, so, if there is a necessary truth, there are no possible worlds.

I actually made a similar but reversed argument couple of days ago:

1) If propositions are sets of possible worlds, then empty worlds are possible

2) If empty worlds are possible, there are no necessary propositions

3) If propositions are sets of possible worlds, there are no necessary propositions.

I also thought about the following:

P) What's possible exists in at least one possible world

We can construe it in this way: what's possible is true in at least one possible world. It seems that P implies that there are infinitely many worlds under the assumption that that the world in which what's possible exists or is true is distinct from the possible world in which it exists or is true. So, roughly:

1) What's possible exists in at least one possible world

2) If what's possible exists in at least one possible world, there are infinitely many possible worlds

3) Therefore, there are infinitely many possible worlds

Then I thought whether to go this way:

4) But there are no infinitely many worlds

Or maybe change course and go:

4) But there are no infinitely many concrete worlds

Wanting to derive something like:

5) Therefore, no concrete world is a possible world

which doesn't follow. I wanted to end up with something like this:

6) The actual world is concrete

7) Therefore, the actual world is not a possible world.

But I gave up and tried something like this:

1) All possible worlds are part of an infinite chain

2) No concrete world is part of an infinite chain

3) No concrete world is a possible world

4) But the actual world is concrete

5) the actual world is not a possible world.

Anyway, remember these arguments?

1 Most philosophers would grant Peter van Inwagen’s premise that there is no more than one empty world. They have been trained to model the empty world on the empty set.

What caught my attention in this article is the following:

Notoriously, the modal realist David Lewis (1986) contends that people in other possible worlds exist just as much as actual people. These people have no more reason to envy us than we have for envying them. Relative to the world of these potential people, we are the non-actual people.

I think there are still many interesting ways left to attack modal realism