There are mathematical structures that allow for comparison but not arithmetic. For instance, a total order on a set is a relation < such that (1) for any x,y we have exactly one of x<y, x=y, y<x and (2) for any x,y,z if x<y<z then x<z. (We say it's trichotomous and transitive.)

The usual ordering on (say) the real numbers is a total order (and it's "compatible with arithmetic" in a useful sense), but there are totally ordered sets that don't look much like any system of numbers.

There are weaker notions of ordering (e.g., a partial order is the same except that condition 1 says "at most one" instead of "exactly one", and allows for some things to be incomparable with others; a preorder allows things to compare equal without being the same object) and stronger ones (e.g., a well-order is a total order with the sometimes-useful property that there's no infinite "descending chain" a1 > a2 > a3 > ... -- this is the property you need for mathematical induction to work).

The ordinals, which some people have mentioned, are a generalization of the non-negative integers, and comparison of ordinals is a well-order, but arithmetic (albeit slightly strange arithmetic) is possible on the ordinals and if you're thinking about preferences then the ordinals aren't likely to be the sort of structure you want.