A common mistake is to take one’s formalism as metaphysics. This is especially true in domains tangentially related to the study of human behavior: “just because you can be described by a coherence theorem does not mean you are a coherence theorem.”
I note that the difference between cardinal and ordinal utilities is not as deep as it may seem. Cardinalists use a utility function \(u: X \to \mathbb{R}\) to describe preferences, while ordinalists restrain themselves to only defining an ordering over \(X.\)
Under natural conditions, orderings over \(X\) can be described as utility functions over \(X.\) If the ordering is complete, transitive, continuous1, and admits an order-dense subset2, there exists a continuous function \(u\) such that \(u(x) \leq u(y)\) if and only if \(x \leq y.\)
As an example: if \(X\) is any convex subset of \(\mathbb{R}^n\) and \(\leq\) is continuous, then this holds and there exists a corresponding continuous utility function.
Give \(X\) topological structure. If the upper and lower contour sets of an ordering \(\leq\) are closed in \(X\) for every \(x \in X,\) then \(\leq\) is continuous.↩︎
There exists \(Z \subseteq X\) such that for every pair \(x,y \in X\) such that \(x \leq y,\) there exists \(z \in Z\) such that \(x \leq z \leq y.\)↩︎