Abstract

This chapter sets out the theorems, and presents some examples that show in a rough way how the theorems work. It explains separability precisely, and states the theorems. The chapter starts the work of interpreting the theorems, and also explains the significance of their conclusions from a formal, mathematical point of view. It then discusses a significant assumption that is used in the proofs of the theorems. The published proofs of both the separability theorems depend on an assumption that may be objectionable in some contexts. These theorems are about an ordering defined on a field of alternatives. If an ordering can be represented by an additively separable utility function, the theorem shows it can be represented by a whole family of additively separable functions. Each member of the family is an increasing linear transform of the others, and the family includes every increasing linear transform of each of its members.