Set.empty is that set; although you can’t get at it directly, it turns out that it is just a private object in the Set companion object (called, obviously enough, EmptySet). All that Set.empty does is return that set with a cast to the correct type.
It is done this way, instead of with Nil, because sets are invariant in their parameters. Nil is List[Nothing](), but you couldn’t add anything to a Set[Nothing]().
If you need to specify the type of your empty set, you can use e.g. Set.empty[String].