Uniform definition of sets using relations and complement of Presburger Arithmetic

TL;DR

This paper demonstrates that for any relation not definable in Presburger Arithmetic, one can define a set of integers using a uniform formula that is not ultimately periodic and can be chosen to be expanding, independent of the relation's interpretation.

Contribution

It introduces a uniform way to define non-FO[N,+,<]-definable sets using relations and complements, independent of the specific relation.

Findings

Existence of a uniform formula for non-definable sets

The formula can define non-ultimately periodic sets

The set can be chosen to be expanding

Abstract

In 1996, Michaux and Villemaire considered integer relations $R$ which are not definable in Presburger Arithmetic. That is, not definable in first-order logic over integers with the addition function and the order relation (FO[N,+,<]-definable relations). They proved that, for each such $R$, there exists a FO[N,+,<,$R$]-formula $\nu_{R}(x)$ which defines a set of integers which is not ultimately periodic, i.e. not FO[N,+,<]-definable. It is proven in this paper that the formula $\nu(x)$ can be chosen such that it does not depend on the interpretation of $R$. It is furthermore proven that $\nu(x)$ can be chosen such that it defines an expanding set. That is, an infinite set of integers such that the distance between two successive elements is not bounded.