Interpretations of Presburger Arithmetic in Itself

  • Alexander ZapryagaevEmail author
  • Fedor Pakhomov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10703)


Presburger arithmetic \(\mathop {\mathbf {PrA}}\nolimits \) is the true theory of natural numbers with addition. We study interpretations of \(\mathop {\mathbf {PrA}}\nolimits \) 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 \((\mathbb {N},+)\) 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 \(\mathop {\mathbf {PrA}}\nolimits \) it follows that \(\mathop {\mathbf {PrA}}\nolimits \) isn’t one-dimensionally interpretable in any of its finite subtheories. We note that the latter was conjectured by A. Visser.


Presburger Arithmetic Interpretations Scattered linear orders 



The authors wish to thank Lev Beklemishev for suggesting to study Visser’s conjecture, number of discussions of the subject, and his useful comments on the paper.


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© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussian Federation

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