I have a very specific question on existence of ordinal utility functions. In many sources (e.g. Krantz et al. 1971: Foundations of measurement: additive and polynomial representations), it is stated that the necessary and sufficient condition on the existence of an ordinal utility function in the general case is that of order separability (i.e., the existence of a countable order-dense subset).
Let $A$ be a set and $\succsim$ be a weak order (a transitive and complete binary relation) on $A$. $\succ$ denotes the corresponding strict relation.
Order separability is defined as follows: There is a countable subset $B$ of $A$ such that for all $x,z \in A$ such that $x \succ z$, there is $y \in B$ such that $x \succsim y \succsim z$.
It is known that a lexicographic ordering does not admit a real-valued utility function, and there are proofs showing that it leads to a contradiction (like the one in Bridges and Mehta), but at least the proofs that I've seen do not show how the lexicographic ordering conflicts with this definition of order separability, or maybe I have been overlooking something. In any case, I would like to see how lexicographic ordering (supposedly on an infinite set) does not satisfy the above definition of order separability, also to understand better how order separability works.
Would somebody be able to help me out here?
Best regards,
Janne
For dictionary ordered R×R look at the intervals ((r,0), (r,1)), r in R.
They form an uncountable collection of not empty, pairwise disjoint sets.
No countable set can have one element in each of those sets.