R-constrain ed R
R
w N
R
Eq

$$\mathbb{N}$$
$$\mathbf{Id}$$
$$\mathrm{d}$$
$$\mathbb{G}$$
$$\mathbf{b}$$

q

## 3. The Counterexample To Cox'S Theorem

Theorem 3.1: There is a function Beb, a finite domain   , and functions  ,  , and satisfying A1, A2, and A3 respectively such that Belo

$$\begin{array}{l l l}{\in{}}&{{}}&{{}f o\,r}\end{array}$$
$\uparrow$  . 
$\left\{h a\right\}$ . 
(so that is strictly decreasing and infinitely differentiable),
+  (so that   is strictly increasing in each argument and is infinitely differentia ble),
is infinitely differentiable, nondecreasing in each argument in       2, and strictly increasing in each argument in 2. Moreover, is commutative,
, and However, there is no one-to-one onto function satisfying (1).

N
0 o p q