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arXiv · 1611.03839

Uniform definition of sets using relations and complement of Presburger Arithmetic

Arthur Milchior
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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 $ν_{R}(x)$ 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 $ν(x)$ can be chosen such that it does not depend on the interpretation of $R$. It is furthermore proven that $ν(x)$ 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.

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Arthur Milchior. 2016-11-11. Uniform definition of sets using relations and complement of Presburger Arithmetic. https://arxiv.org/abs/1611.03839

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