even if the zero gets moved arbitrarily to another point, or if the choice of scale is changed, or if both the scale and the zero are changed. Every measurable entity maps into a cardinal function but not every cardinal function is the result of the mapping of a measurable entity. The point of this example was used to prove that (as with temperature) it is still possible to predict something about the combination of two values of some utility function, even if the utils get transformed into entirely different numbers, as long as it remains a linear transformation. Von Neumann and Morgenstern stated that the question of measurability of physical quantities was dynamic. For instance, temperature was originally a number only up to any monotone transformation, but the development of the ideal-gas-thermometry led to transformations in which the absolute zero and absolute unit were missing. Subsequent developments of thermodynamics even fixed the absolute zero so that the transformation system in thermodynamics consists only of the multiplication by constants. According to Von Neumann and Morgenstern (1944, p. 23) "For utility the situation seems to be of a similar nature [to temperature]". The following quote from Alchian served to clarify once and for all the real nature of utility functions, emphasizing that they no longer need to be measurable: Order of preference In 1955 Patrick Suppes and Muriel Winet solved the issue of the representability of preferences by a cardinal utility function, and derived the set of axioms and primitive characteristics required for this utility index to work. Suppose an agent is asked to rank his preferences of and his preferences of . If he finds that he can state, for example, that his degree of preference of exceeds his degree of preference of , we could summarize this information by any triplet of numbers satisfying the two inequalities: and . If and were sums of money, the agent could vary the sum of money represented by until he could tell us that he found his degree of preference of over the revised amount equal to his degree of preference of over . If he finds such a {{math|B'''}}, then the results of this last operation would be expressed by any triplet of numbers satisfying the relationships: (a) , and (b) = . Any two triplets obeying these relationships must be related by a linear transformation; they represent utility indices differing