Utility functions on ordered convex sets

The work of Cantor in set theory near the end of the last century showed that a linearly ordered set need not be order isomorphic to a subset of the real numbers. To obtain order isomorphism, it is necessary and sufficient that some countable subset of the linearly ordered set be order- dense in the entire set. The present paper proves a negative result that is much stronger than Cantor's. It shows that a weakly ordered convex set whose order relation is ‘continuous’ and ‘totally convex’ need not be order homomorphic to a subset of the reals. To obtain order homomorphism, it is necessary and sufficient that the weakly ordered set be countably bounded, i.e., have countable coinitial and cofinal subsets. Countable boundedness is a significantly weaker condition than countable order-denseness. Connections between these results and economic utility theory are discussed.