Condorcet-Style Paradoxes for Majority Rule with Infinite Candidates

I present two possibility results and one impossibility result about a situation with three voters under a pairwise majoritarian aggregation function voting on a countably infinite number of candidates. First, from individual orders with no maximal or minimal element, it is possible to generate an aggregate order with a maximal or minimal element. Second, from dense individual orders, it is possible to generate a discrete aggregate order. Finally, I show that, from discrete orders with a particular property, namely the finite-distance property, it is not possible to generate a dense aggregate order.