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Appears in collection : XVII International Luminy Workshop in Set Theory / XVII Atelier International de Théorie des Ensembles

The purpose of this article is to give new constructions of linear orders which are minimal with respect to being non-σ-scattered. Specifically, we will show that Jensen's principle ♢ implies that there is a minimal Countryman line, answering a question of Baumgartner. We also produce the first consistent examples of minimal non-σ-scattered linear orders of cardinality greater than ℵ1, as given a successor cardinal κ+, we obtain such linear orderings of cardinality κ+ with the additional property that their square is the union of κ-many chains. We give two constructions: directly building such examples using forcing, and also deriving their existence from combinatorial principles. The latter approach shows that such minimal non-σ-scattered linear orders of cardinality κ+ exist for every cardinal κ in Gödel's constructible universe, and also (using work of Rinot) that examples must exist at successors of singular strong limit cardinals in the absence of inner models satisfying the existence of a measurable cardinal μ of Mitchell order μ++.

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Citation data

  • DOI 10.24350/CIRM.V.20105403
  • Cite this video Moore Justin (10/10/23). Large minimal non-sigma-scattered linear orders. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20105403
  • URL https://dx.doi.org/10.24350/CIRM.V.20105403



  • EISWORTH, Todd, CUMMINGS, James, et MOORE, Justin Tatch. On minimal non-$\sigma $-scattered linear orders. arXiv preprint arXiv:2304.03389, 2023. - https://arxiv.org/abs/2304.03389

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