There have been misunderstandings. Some think U requires of social welfare functions that they can handle “any old” alternatives. It does nothing of the sort. What it requires is that the social welfare function can handle the widest possible range of preferences among whichever alternatives there are to choose among, and whether there happen to be many of these or only a few of them is beside the point: the domain of a social welfare function can be completely unrestricted even if there are in \(X\) just two alternatives. One way to sustain this unorthodox understanding of U is, perhaps, to think of Arrow’s \(x, y, z, \ldots\) not as alternatives properly speaking—not as candidates in elections, social states, or what have you—but as names or labels that represent these on different occasions for choosing. Then, it might be thought, variety among the alternatives to which the labels can be attached will generate variety among the profiles that an aggregation procedure might be expected to handle.  Blackorby et al. (2006) toy with this idea at one point, but they quickly set it aside. It does not seem to have been explored in the literature.