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Topological models of arithmetic

Volume 256 / 2022

Ali Enayat, Joel David Hamkins, Bartosz Wcisło Fundamenta Mathematicae 256 (2022), 171-193 MSC: Primary 03C62; Secondary 03C66. DOI: 10.4064/fm928-1-2021 Published online: 10 May 2021

Abstract

Ali Enayat had asked whether there is a model of PA (Peano Arithmetic) that can be represented as $\langle \mathbb Q ,\oplus ,\otimes \rangle $, where $\oplus $ and $\otimes $ are continuous functions on the rationals $\mathbb Q $. We prove, affirmatively, that indeed every countable model of PA has such a continuous presentation on the rationals. More generally, we investigate the topological spaces that arise as such topological models of arithmetic. Finite-dimensional Euclidean spaces $\mathbb R ^n$ and compact Hausdorff spaces do not, and neither does any Suslin line; many other spaces do. The status of the space of irrationals remains open.

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