Spacetime and the Worldmate Relation
Lewis does not want to take the worldmate relation (that holds between two things iff they belong to the same world) as primitive. He proposes two alternatives. The first is that things belong to the same world iff they stand in ("analogously") spatiotemporal relations to each other. According to the second, more general, proposal things belong to the same world iff they stand in fundamental external relations to each other, whether or not these relatios are (analogously) spatiotemporal. I'm not sure if I fully understand the difference between these three alternatives. Here is why.
Lewis agrees that there are five-dimensional worlds. In particular, there is a five-dimensional world some of whose four-dimensional slices pretty much resemble our spacetime (see Plurality, p.72). But why is this a single world and not many -- many four-dimensional worlds taken together? What makes the relations along the fifth dimension spatiotemporal? (At least on Lewis' account, appealing to laws of physics won't help.) Maybe Lewis would reply that this questions can't be answered, for spatiotemporal relations are fundamental relations, and if it is a fundamental fact that x is R-related to y, then one can't expect there to be a good explanation of what makes it the case that x is R-related to y.
So there are all kinds of four-dimensionally maximal fusions (let's call them 'spacetimes'). Some of them are complete worlds, like perhaps our world. Others are slices of bigger worlds, together with further spacetimes. And this is a primitive fact. There is nothing about the instrinsic nature of a spacetime that could tell you whether it is a world on its own. Nor is there anything about the intrinsic nature of two spacetimes that could tell you whether they belong to the same world.
The situation carries over to smaller things. We have a more or less unproblematic sufficient condition: If two things are related to each other in exactly the same way that all parts of our spacetime are related to each other then they are worldmates. But what about things that are not so related? They still might be worldmates. Or maybe they aren't. Whether they are depends on whether they stand to each other in some inexplicable fundamental relation. In what sense is this different from taking the worldmate relation as primitive?