Nope -- unfortunately, the voting analogy is a poor way to understand ultrafilters. Essentially, filters (a dual to ideals) are structures that tell you which

Nope -- unfortunately, the voting analogy is a poor way to understand ultrafilters. Essentially, filters (a dual to ideals) are structures that tell you which subsets of a partially ordered set are "big", and ultrafilters are maximal filters. A pathological example that can help illuminate the idea is that of a principal ultrafilter ("on x"), which is all subsets of a partially ordered set containing x as a least element. The elements of the ultrafilter are the majorities, and their complements the minorities -- clearly the union of two sets not containing x will not contain x, and the intersection of two sets containing x will always contain x. Choice gives you the existence of a nonprincipal ultrafilter, that is, a structure on the integers without a specific least element which correctly identifies all "big" subsets of the integers in a way that provides that the intersection of "big" sets is still "big". I'd highly recommend the treatment in Just and Weese if you're interested in learning more.