Topological models of arithmetic
Volume 256 / 2022
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.
