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Preprint Number 1890

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1890. Alfred Dolich, Chris Miller, Alex Savatovsky, Athipat Thamrongthanyalak
Connectedness in structures on the real numbers: o-minimality and undecidability

Submission date: 30 November 2020


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-minimal structures on (ℝ,<) have the property, as do all expansions of (ℝ,+,×,ℕ). Our main analytic-geometric result is that any such expansion of (ℝ,<,+) by boolean combinations of open sets (of any arities) either is o-minimal or defines an isomorph of (ℕ,+,×). We also show that any given expansion of (ℝ, <, +,ℕ) by subsets of 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.

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Full text arXiv 2011.14833: pdf, ps.

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