Topological models of arithmetic

2018/08/03 by Enayat, Ali, Hamkins, Joel David, Wcisło, Bartosz
#03H15 (Primary) #54H10 (Secondary) #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1808.01270

Abstract

Ali Enayat had asked whether there is a nonstandard model of Peano arithmetic (PA) that can be represented as ⟨ℚ,⊕,⊗⟩, where ⊕ and ⊗ 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 ℝ, the reals in any finite dimension ℝⁿ, 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.

Related

• On Non-Standard Models of Peano Arithmetic and Tennenbaum's Theorem
• Uncountable Real Closed Fields with PA Integer Parts
• A Topological Tennenbaum Theorem
• Classifying the complexity of models of arithmetic
• The structural complexity of models of arithmetic