# Continuous models of arithmetic, MOPA, November 2020

This will be a talk for the Models of Peano Arithmetic (MOPA) seminar on 11 November 2020, 12 pm EST (5pm GMT). Kindly note the rescheduled date and time.

Abstract. Ali Enayat had asked whether there is a model of Peano arithmetic (PA) that can be represented as $\newcommand\Q{\mathbb{Q}}\langle\Q,\oplus,\otimes\rangle$, where $\oplus$ and $\otimes$ are continuous functions on the rationals $\Q$. 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.

This is joint work with Ali Enayat, myself and Bartosz Wcisło.

Article: Topological models of arithmetic