Differentiating Infinite Voting Populations using Ultrafilters

Ultrafilters arise frequently in the social choice literature and surrounding fields as collections of the decisive set for a given aggregation procedure. We know from [8] that we can associate each Arrow social welfare function with an ultrafilter. Because of their structure, ultrafilters are useful in analyses of both finite and infinite voting populations. For an infinite set of voters, [5] demonstrated that given a society with an infinite population, the non-dictatorship condition of Arrow’s general possibility theorem is satisfied, in that there is no single voter whose preferences dictate the outcome of the election. Similar results regarding non-dictatorship in infinite populations have been proven in the fields of judgment aggregation as well [4]. Less attention has been paid to the study of the comparison of societies with infinite populations. One of the first efforts to do so was made by [12], utilizing the Rudin-Keisler order over ultrafilters. After this, however, little further work has been undertaken. In this paper, we focus on the study of the relationship between societies with countably infinite voters. Thus, for a given population, the cardinality of the population of voters in our society is equal to the cardinality of the natural numbers: |V | = |N|. In this process of comparison, a natural question to ask would be if all sets of decisive voters (ultrafilters) over N are isomorphic, and thus fundamentally similar. To answer this, we present the following theorem due to [10]: