I think you've missed the point that willun is making. It's in general impossible to assign a numeric "rank" to a partially ordered set such that the rank(A) >

I think you've missed the point that willun is making. It's in general impossible to assign a numeric "rank" to a partially ordered set such that the rank(A) > rank(B) iff A > B and rank(A) = rank(B) iff A is incomparable to B. I'm pointing out a concrete example where you cannot assign a rank to France that is consistent with the rank of Italy and the rank of Korea. Since France is incomparable to both Italy and Korea, it must have the same rank as both, but Korea is better ranked than Italy. Contradiction.