Multi-dimensional Interpretations of Presburger Arithmetic in Itself

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.

TWO NEW SERIES OF PRINCIPLES IN THE INTERPRETABILITY LOGIC OF ALL REASONABLE ARITHMETICAL THEORIES

The provability logic of a theory T captures the structural behavior of formalized provability in T as provable in T itself. Like provability, one can formalize the notion of relative interpretability giving rise to interpretability logics. Where provability logics are the same for all moderately sound theories of some minimal strength, interpretability logics do show variations.The logic IL (All) is defined as the collection of modal principles that are provable in any moderately sound theory of some minimal strength. In this article we raise the previously known lower bound of IL (All) by exhibiting two series of principles which are shown to be provable in any such theory. Moreover, we compute the collection of frame conditions for both series.

Decidability of interpretability logics ILM0 and ILW*

The finite model property is a key step in proving decidability of modal logics. By adapting the filtration method to the generalized Veltman semantics for interpretability logics, we have been able to prove the finite model property of interpretability logic ILM0 w.r.t. generalized Veltman models. We use the same technique to prove the finite model property of interpretability logic ILW* w.r.t. generalized Veltman models. The missing link needed to prove the decidability of ILM0 was completeness w.r.t. generalized Veltman models, which we obtain in this article. Thus, we prove the decidability of ILM0, which was an open problem. Using the same technique, we prove that ILW* is also decidable.