Let P be a (strict) preference relation over a given set X. A preference relation is a binary relation (i.e. it relates two elements of a given set) that has the properties of asymmetry and negative transitivity.
More specifically, asymmetry means that if x strictly preferred to y (from here on xPy) then y is not preferred to x (from here on x~Py).
Negative transitivity means that if x~Py & y~Pz => x~Pz. Translating that means that if x not preferred to y and y not preferred to z, then x is not preferred to z.
These sound like reasonable axioms, don't you think?
First homework, prove that the properties of asymmetry and negative transitivity over strict preferences imply completeness and transitivity of the weak preferences. Note: weak preference of x over y just means that x is strictly preferred or indifferent to y.
I'll continue as soon as someone proves this. The next step will be to prove that there is a unique (up to affine transformations) function u for which xPy => u(x) > u(y), for all pairs (x,y) in X.