Interpretations of Presburger Arithmetic in Itself

Alexander Zapryagaev; Fedor Pakhomov

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arXiv · 1709.07341

Interpretations of Presburger Arithmetic in Itself

Alexander Zapryagaev·Fedor Pakhomov

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Abstract

Presburger arithmetic PrA is the true theory of natural numbers with addition. We study interpretations of PrA 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 (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 PrA it follows that PrA isn't one-dimensionally interpretable in any of its finite subtheories. We note that the latter was conjectured by A. Visser.

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BibTeX RIS

Alexander Zapryagaev, Fedor Pakhomov. 2017-09-21. Interpretations of Presburger Arithmetic in Itself. https://arxiv.org/abs/1709.07341

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