Unique Finite Difference Measurement

This paper considers a variety of combinatorial and number-theoretic questions that arise from considerations of uniqueness in the theory of measurement. Given $n + 1$ objects linearly ordered from worst (smallest) to best (largest), let $d_i $ stand for the difference between objects i and $i + 1$. Three types of assumptions that allow different inequality and equality comparisons between certain subsets of differences are considered. The three types of assumptions arise from problems concerning the uniqueness of finite algebraic difference measurement and finite subjective probability measurement. If the number of equality comparisons is sufficient to imply that the $d_i $ in $d = ( d_1 ,d_2 , \cdots ,d_n )$ are unique up to multiplication by a positive constant, then d is said to be unique. Results on the number of unique d’s and relationships among the components of such d’s are obtained for each of the three types.