A function is a utility function if it satisfies: u:X -> R x >= y <=> u(x) => u(y) uii=u(xii) Existence if => a weak order: Cantor's theorem in 1895 says: => a weak order on X /\ X/~ (the equivalence classes) at most numerable => there exists u:X->R s.t. forall x,y in X x=>y <=> u(x)=>u(y) this is later extended to: => a weak order on X /\ X/~ (the equivalence classes) has a dense numerable subset=> there exists u:X->R s.t. forall x,y in X x=>y <=> u(x)=>u(y) Utility can be measured with a randomizing device. let w in O = states of the world X= a set of consequences F= a set of acts f:O->X Let O be finite => 2^O is an algebra. Find a probability charge P on (O,2^O) Find a utility function u:X->R s.t. f=>g <=> Sum(w in O,u(f(w))p(w))=>Sum(w in O,u(g(w))p(w)) The construction: There exists E Subset of O which is ethically neutral. There exists x">"y,x'">"y' (x,E;y,Ec)=>(x',E;y',Ec) <=> u(x)-u(x')=>u(y)-u(y') Method: find the smallest and largest X (xl,xu) , assign them u(x)=0,1 respectively. let u(xl,E;xu,Ec)=1/2... build in from there.