Connectedness in structures on the real numbers: o-minimality and undecidability

TL;DR

This paper explores structures on real numbers where definable sets have definable path components, revealing that such structures are either o-minimal or encode the complexity of arithmetic, with implications for decidability.

Contribution

It characterizes expansions of real ordered structures with definable path components, showing they are either o-minimal or encode arithmetic, and analyzes their decidability properties.

Findings

Expansions of $(R,<,+)$ by open sets are either o-minimal or encode $(N,+, imes)$.

Expansions of $(R,<,+,N)$ by subsets of $N^n$ define all arithmetic sets.

Connected components and quasicomponents lead to similar dichotomies.

Abstract

We initiate an investigation of structures on the set of real numbers having the property that path components of definable sets are definable. All o\nobreakdash-\hspace{0pt}minimal structures on $(\mathbb{R},<)$ have the property, as do all expansions of $(\mathbb{R},+,\cdot,\mathbb{N})$. Our main analytic-geometric result is that any such expansion of $(\mathbb{R},<,+)$ by boolean combinations of open sets (of any arities) either is o\nobreakdash-\hspace{0pt}minimal or defines an isomorph of $(\mathbb N,+,\cdot\,)$. We also show that any given expansion of $(\mathbb{R}, <, +,\mathbb{N})$ by subsets of $\mathbb{N}^n$ ($n$ allowed to vary) has the property if and only if it defines all arithmetic sets. Variations arise by considering connected components or quasicomponents instead of path components.