We consider the problem of ranking sets of alternatives. Standard approaches to this problem regard the addition of an alternative to a set containing one element as enhancing choice. We argue that this monotonicity axiom may not be desirable when an agent is uncertain as to the value of this additional alternative. We replace monotonicity with an uncertainty aversion axiom, and also introduce an axiom that produces lexicographic behaviour. These axioms, in conjunction with an independence axiom, enable us to prove a characterisation theorem. This theorem says that sets are ranked in terms of the number of uncertain elements that they contain, the fewer the better. This is the only ranking rule that satisfies our axioms.