 (iii) It could help us decide whether some list of equational laws intended to axiomatize the integers is complete in the sense that any equation holding of the integers follows from the laws in that list. If some structure satisfies all the axioms in the list, but not some other equation that holds of the integers, then we have a witness to the incompleteness of the axiomatization. If on the other hand we can show how to construct any algebra satisfying the axioms from the algebra of integers, limiting ourselves only to certain algebraic constructions, then by a theorem of Birkhoff applicable to those constructions we can infer that the axiomatization is complete.