I think this paper is refuting Conway (and others') proof of the claim that a set can be divided into 3+ parts without relying on the Axiom of Choice. It does so by observing that the proof only applies to sets that have an ordering, which is to say that there is an algorithm for picking an element from the set, which am extra constraint beyond having only a test for determining whether a known object is an element is in the set. It then goes on to prove that it is not possible to evenly divide a set that lacks such a selection algorithm.