Multiset repetitiveness

Once we accept that our pure set theory must be enriched with multisets which can be produced from any set system by combining sets counting multiplicity, then it is interesting to consider how sets fit into the picture of multiset theory. Given any multiset we can get the distance from that multiset to a set under the multiset ordering, which is essentially the repetitiveness of the multiset.

(defn repetitiveness
  [coll]

  (- (count (multiset coll))
     (count (set coll))))

This means that sets are essentially repetitiveness zero multisets. More interesting though is the concept of near-sets which are multisets which have no more then a distance of one from a set. So the most set like multisets are sets themselves, then near sets, then near near sets, and so on. As you can see, there is a hierarchy of different degrees of setness in multiset theory.