On models of affine arithmetic

TL;DR

This paper explores the properties of affine arithmetic, a continuous and undecidable extension of Peano arithmetic, analyzing its models within affine logic and establishing affine analogs of classical results like Gaifman's theorem.

Contribution

It introduces affine variants of classical Peano arithmetic results and characterizes models of affine arithmetic within the framework of affine logic.

Findings

Affine arithmetic is undecidable.

Models are generally lattice ordered with a nontrivial metric.

Affine variants of classical results, including Gaifman's splitting theorem, are established.

Abstract

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.