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H 1 Lower bounds for the perfect set property at weakly compact cardinals

Auteurs : Müller, Sandra (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : By the Cantor-Bendixson theorem, subtrees of the binary tree on $\omega$ satisfy a dichotomy - either the tree has countably many branches or there is a perfect subtree (and in particular, the tree has continuum manybranches, regardless of the size of the continuum). We generalize this to arbitrary regular cardinals $\kappa$ and ask whether every $\kappa$-tree with more than $\kappa$ branches has a perfect subset. From large cardinals, this statement isconsistent at a weakly compact cardinal $\kappa$. We show using stacking mice that the existence of a non-domestic mouse (which yields a model with a proper class of Woodin cardinals and strong cardinals) is a lower bound. Moreover, we study variants of this statement involving sealed trees, i.e. trees with the property that their set of branches cannot be changed by certain forcings, and obtain lower bounds for these as well. This is joint work with Yair Hayut.

    Keywords : immer models, core models, stacking mice, weakly compact cardinal, $\kappa$-trees

    Codes MSC :
    03E05 - Combinatorial set theory (logic)
    03E35 - Consistency and independence results
    03E45 - Constructibility, ordinal definability, and related notions
    03E55 - Large cardinals

    Ressources complémentaires :
    https://www.cirm-math.fr/RepOrga/2052/Slides/MuellerLuminy.pdf

      Informations sur la Vidéo

      Réalisateur : Hennenfent, Guillaume
      Langue : Anglais
      Date de publication : 14/10/2019
      Date de captation : 23/09/2019
      Collection : Research talks
      Format : MP4
      Durée : 00:33:12
      Domaine : Logic and Foundations
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2019-09-23_Muller.mp4

    Informations sur la rencontre

    Nom de la rencontre : 15th International Luminy Workshop in Set Theory / XVe Atelier international de théorie des ensembles
    Organisateurs de la rencontre : Dzamonja, Mirna ; Velickovic, Boban
    Dates : 23/09/2019 - 27/09/2019
    Année de la rencontre : 2019
    URL Congrès : https://conferences.cirm-math.fr/2052.html

    Citation Data

    DOI : 10.24350/CIRM.V.19564003
    Cite this video as: Müller, Sandra (2019). Lower bounds for the perfect set property at weakly compact cardinals. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19564003
    URI : http://dx.doi.org/10.24350/CIRM.V.19564003


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