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- W3108170844 abstract "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-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-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." @default.
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- W3108170844 date "2022-02-18" @default.
- W3108170844 modified "2023-09-27" @default.
- W3108170844 title "CONNECTEDNESS IN STRUCTURES ON THE REAL NUMBERS: O-MINIMALITY AND UNDECIDABILITY" @default.
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- W3108170844 doi "https://doi.org/10.1017/jsl.2022.16" @default.
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