Interpretations of Presburger Arithmetic in Itself

TL;DR

This paper investigates how Presburger arithmetic interprets itself, showing that all one-dimensional self-interpretations are essentially the same as the identity, and exploring the structure of interpretable linear orders.

Contribution

It proves that all one-dimensional self-interpretations of Presburger arithmetic are definably isomorphic to the identity, and characterizes interpretable linear orders as scattered with bounded Hausdorff rank.

Findings

All one-dimensional self-interpretations are isomorphic to the identity.

Interpretable linear orders are scattered with finite Hausdorff rank.

Presburger arithmetic is not one-dimensionally interpretable in its finite subtheories.

Abstract

Presburger arithmetic PrA is the true theory of natural numbers with addition. We study interpretations of PrA in itself. We prove that all one-dimensional self-interpretations are definably isomorphic to the identity self-interpretation. In order to prove the results we show that all linear orders that are interpretable in (N,+) are scattered orders with the finite Hausdorff rank and that the ranks are bounded in terms of the dimension of the respective interpretations. From our result about self-interpretations of PrA it follows that PrA isn't one-dimensionally interpretable in any of its finite subtheories. We note that the latter was conjectured by A. Visser.