  Arrow's `impossibility' theorem asserts that there are no satisfactory
methods of aggregating individual preferences into collective preferences in
many complex situations. This result has ramifications in economics, politics,
i.e., the theory of voting, and the structure of tournaments. By identifying
the objects of choice with mathematical sets, and preferences with Hausdorff
measures of the distances between sets, it is possible to extend Arrow's
arguments from a sociological to a mathematical setting. One consequence is
that notions of reversibility can be expressed in terms of the relative
configurations of patterns of sets.
