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

Type of Publication: | Journal article |

Publication status: | Published |

Author: | Dolich, Alfred; Miller, Chris; Savatovsky, Alex; Thamrongthanyalak, Athipat |

Year of publication: | 2022 |

Published in: | 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 |

DOI (citable link): | https://dx.doi.org/10.1017/jsl.2022.16 |

Summary: |
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. |

Subject (DDC): | 510 Mathematics |

Refereed: | Yes |

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DOLICH, 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

@article{Dolich2022CONNE-58617, title={CONNECTEDNESS IN STRUCTURES ON THE REAL NUMBERS: O-MINIMALITY AND UNDECIDABILITY}, year={2022}, doi={10.1017/jsl.2022.16}, 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} }