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Preprint Number 1752

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1752. Fedor Pakhomov and Alexander Zapryagaev
Multi-Dimensional Interpretations of Presburger Arithmetic in Itself
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Submission date: 5 April 2020

Abstract:

Presburger Arithmetic is the true theory of natural numbers with addition. We study interpretations of Presburger Arithmetic in itself. The main result of this paper is that all self-interpretations are definably isomorphic to the trivial one. Here we consider interpretations that might be multi-dimensional. We note that this resolves a conjecture by A. Visser. In order to prove the result we show that all linear orderings that are interpretable in (ℕ;+) are scattered orderings with the finite Hausdorff rank and that the ranks are bounded in the terms of the dimensions of the respective interpretations.

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Full text arXiv 2004.03404: pdf, ps.


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