Topological models of arithmetic

TL;DR

This paper proves that every countable model of Peano arithmetic can be represented as a continuous structure on the rationals, and explores which topological spaces can serve as such models.

Contribution

It establishes the existence of continuous representations of all countable models of PA on the rationals and characterizes which topological spaces can serve as models.

Findings

Every countable model of PA has a continuous presentation on the rationals.

Certain topological spaces like the reals, long line, and Cantor space cannot serve as such models.

The status of the Baire space as a model remains open.

Abstract

Ali Enayat had asked whether there is a nonstandard model of Peano arithmetic (PA) 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. The reals $\mathbb{R}$, the reals in any finite dimension $\mathbb{R}^n$, the long line and the Cantor space do not, and neither does any Suslin line; many other spaces do; the status of the Baire space is open.