On models of affine arithmetic

By affine arithmetic is meant the set of affine consequences of Peano arithmetic. This is a continuous theory which is studied in the framework of affine logic, a sublogic of continuous logic. Affine arithmetic is undecidable. Also, its models are generally lattice ordered and carry a nontrivial metric. Classical models are then characterized as those which are linearly ordered. In this paper, the affine variants of several classical results in Peano arithmetic are proved. In particular, an affine form of Gaifman's splitting theorem is proved.