You could (using "is a subset" as a partial order), but you can't make a total order. Any way you try to compare size of sets where neither is subset of the oth

You could (using "is a subset" as a partial order), but you can't make a total order. Any way you try to compare size of sets where neither is subset of the other, while preserving your intuition of "size" will run into trouble. You make "the ordinals" sort of using your idea, but that isn't really measurement of "size"; it's more like an assignment of ranks.