Cardinal Utility

Cardinal utility means that utility differences are meaningful. What does meaningful mean?

Suppose we have a set $X$ of objects. A relation is a set $R$ of $n$-tuples of elements of $X$. The order of $R$ is $n$. For instance a preference relation is a set of pairs of elements of $X$; it's order is 2. A relational structure $\mathcal X=\langle X,R_1,R_2,\ldots,R_K\rangle$ contains the set $X$ of objects together with $K$ relations, of order $o_1,o_2,\ldots,o_K$, respectively. We measure $\mathcal X$ by a relational structure $\langle\R,R'_1,\ldots,R'_K\rangle$ where each $R'_k$ is of order $o_k$, and a function $f:X\to\R$ such that $(x_1,\ldots,x_{o_k})\in R_k$ iff $\bigl(f(x_1),\ldots,f(x_{o_k})\bigr)\in R'_k$. The function $f$ is called a scale. For instance, if $\succ$ is a preference relation on $X$, the relational structure is $\langle X,\succ\rangle$, the relational structure on $\R$ is $\langle \R>\rangle$, and the scale is a utility function $u:X\to\R$ that represents $\succ$.