cardinal and ordinal utilities

February 04, 2026

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.