Uniform definition of sets using relations and complement of Presburger Arithmetic

Arthur Milchior

Uniform definition of sets using relations and complement of Presburger Arithmetic

Logic 2016-11-14 v1

Authors: Arthur Milchior

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.

Keywords

Presburger arithmetic integer relations first-order logic uniform definition expanding set ultimately periodic set

Cite

@article{arxiv.1611.03839,
  title  = {Uniform definition of sets using relations and complement of Presburger Arithmetic},
  author = {Arthur Milchior},
  journal= {arXiv preprint arXiv:1611.03839},
  year   = {2016}
}

Uniform definition of sets using relations and complement of Presburger Arithmetic

Abstract

Keywords

Cite

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