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 Visser (1998, An overview of interpretability logic. Advances in Modal Logic, pp. 307–359). In order to prove the result, we show that all linear orderings that are interpretable in $({\mathbb{N}},+)$ are scattered orderings with the finite Hausdorff rank and that the ranks are bounded in the terms of the dimensions of the respective interpretations.

中文翻译：

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Presburger算法本身的多维解释

Presburger算术是真正的自然数加法理论。我们本身研究Presburger算术的解释。本文的主要结果是，所有的自我解释对于琐碎的自我解释都是同构的。在这里，我们考虑可能是多维的解释。我们注意到，这解决了Visser的一个猜想（1998年，可解释性逻辑概述。模态逻辑进展，第307-359页）。为了证明结果，我们显示了在| $（{\ mathbb {N}}，+）$ |中可以解释的所有线性排序。是具有有限Hausdorff等级的分散排序，并且等级受相应解释的维度限制。