On models of affine arithmetic
Seyed-Mohammad Bagheri
TL;DR
This paper develops affine arithmetic (AA), the affine fragment of Peano arithmetic within affine logic, and studies its model theory, undecidability, and number-theoretic content. It shows that AA is undecidable in affine logic, yet inherits a robust toolkit from classical PA, including an affine induction framework, an affine Gaifman splitting theorem, and affine analogues of overspill/collection, Bézout, and division, all within Bauer-theory scenery where PA arises as the extremal models. A key contribution is the affine reduction of PA and the demonstration that AA_{af} forms a Bauer theory whose extremal models are exactly PA models, together with a rich hierarchy (Sigma_n, Pi_n) and a splitting mechanism that parallels Gaifman’s classical results. The work establishes foundational affine-model-theoretic methods—ultramean constructions, end-sets, definability, and saturation—needed to transfer classical arithmetic results into the affine continuous-logic setting, with implications for the expressiveness and limits of affine theories. Overall, the paper extends Peano arithmetic into a continuous-affine framework, proving deep structural theorems (like the affine splitting theorem) and delineating undecidability and definability phenomena in AA.
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.