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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 )

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 : Logic and Foundations Sous collection : Research talks Format : MP4 Domaine : Logic and Foundations Durée : 00:33:12 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 ensemblesOrganisateurs de la rencontre : Dzamonja, Mirna ; Velickovic, BobanDates : 23/09/2019 - 27/09/2019 Année de la rencontre : 2019 URL Congrès : https://conferences.cirm-math.fr/2052.htmlCitation 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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