What are the forms of math called where you can compare numbers, such as to say that 3 is bigger than 2, but can’t necessarily add numbers—that is, 2+2 may or may not be bigger than 3?
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.
As banx said, take ordinal numbers (and remove ordinal addition). Classical ordinal numbers can be added, but they can’t be scaled—you can’t generally have x% of an ordinal number.
What are the forms of math called where you can compare numbers, such as to say that 3 is bigger than 2, but can’t necessarily add numbers—that is, 2+2 may or may not be bigger than 3?
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.
I think you’re talking about ordinal numbers vs cardinal numbers. With ordinal numbers you can say a > b, but not by how much.
As banx said, take ordinal numbers (and remove ordinal addition). Classical ordinal numbers can be added, but they can’t be scaled—you can’t generally have x% of an ordinal number.