Question

Suppose we have a non-strict total ordering on tuples of real numbers (the ordering includes tuples of differing lengths). Any tuple, t2, generated from another tuple, t1, by increasing one or more values has t2>t1. Is there necessarily a function f from tuples to real numbers which preserves the ordering? Can this be easily further generalised?

Note:

• By non-strict, I mean that different elements may be equivalent
• This question is actually motivated by investigating Utilitarianism. The tuples are tuples of individual utilities and the aim is to see whether the ability to rank tuples of utilities (with one common sense restriction on the ordering) necessarily means that that a combined utility function can be created

Answer 1

I'm not sure what the point of taking tuples is, since all you seem to care about is the cardinality of the set of tuples (which is the same as the cardinality of the reals).

If you're asking whether every total ordering on a set with the cardinality of the reals is order-isomorphic to the usual ordering on a subset of the reals, the answer is no. In the reals, there can be at most countably many disjoint intervals, but that's not true in the long line.

If you're asking whether tuples with the lexicographic total order are order-isomorphic to a subset of the reals, the answer is again no, even for 2-tuples, for the same reason: the 2-tuples have uncountably many disjoint intervals of the form (a,b)-(a,c) (with b < c).

As for "I'm not really sure what area of maths this is": order theory.