CONNECTEDNESS IN STRUCTURES ON THE REAL NUMBERS: O-MINIMALITY AND UNDECIDABILITY

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2022
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Dolich, Alfred
Miller, Chris
Thamrongthanyalak, Athipat
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10.1017/jsl.2022.16
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The Journal of Symbolic Logic ; 87 (2022), 3. - pp. 1243-1259. - Cambridge University Press on behalf of the Association for Symbolic Logic. - ISSN 0022-4812. - eISSN 1943-5886
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 (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 Nn (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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ISO 690DOLICH, Alfred, Chris MILLER, Alex SAVATOVSKY, Athipat THAMRONGTHANYALAK, 2022. CONNECTEDNESS IN STRUCTURES ON THE REAL NUMBERS: O-MINIMALITY AND UNDECIDABILITY. In: The Journal of Symbolic Logic. Cambridge University Press on behalf of the Association for Symbolic Logic. 87(3), pp. 1243-1259. ISSN 0022-4812. eISSN 1943-5886. Available under: doi: 10.1017/jsl.2022.16
BibTex
@article{Dolich2022CONNE-58617,
  year={2022},
  doi={10.1017/jsl.2022.16},
  title={CONNECTEDNESS IN STRUCTURES ON THE REAL NUMBERS: O-MINIMALITY AND UNDECIDABILITY},
  number={3},
  volume={87},
  issn={0022-4812},
  journal={The Journal of Symbolic Logic},
  pages={1243--1259},
  author={Dolich, Alfred and Miller, Chris and Savatovsky, Alex and Thamrongthanyalak, Athipat}
}
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