# Séminaires : Séminaire Général de Logique

 Equipe(s) : lm, Responsables : S. Anscombe, O. Finkel, A. Khélif, A. Vignati Email des responsables : sylvy.anscombe@imj-prg.fr, vignati@imj-prg.fr Salle : 1013 Adresse : Sophie Germain Description

 Orateur(s) Dima Sinapova - University of Illinois at Chicago, Titre Iteration, reflection, and singular cardinals Date 21/10/2020 Horaire 17:00 à 18:00 Diffusion https://u-paris.zoom.us/rec/share/izvM4602oLSbb5B6zG4vk2aO1ohrbbXy9mbNs6GFmBOxxQrmybIJsGI3wtnII33b.MEazLEYGXR5ecUnj Résume There is an inherent tension between stationary reflection and the failure of the singular cardinal hypothesis (SCH). The former is a compactness type principle that follows from large cardinals. Compactness is the phenomenon where if a certain property holds for every smaller substructure of an object, then it holds for the entire object. In contrast, failure of SCH is an instance of incompactness. Two classical results of Magidor are: (1) from large cardinals it is consistent to have reflection at $\aleph_{\omega+1}$, and (2) from large cardinals it is consistent to have the failure of SCH at $\aleph_\omega$. As these principles are at odds with each other, the natural question is whether we can have both. We show the answer is yes. We describe a Prikry style iteration, and use it to force stationary reflection in the presence of not SCH. Then we obtain this situation at $\aleph_\omega$ by interleaving collapses. This is joint work with Alejandro Poveda and Assaf Rinot. Salle 1013 Adresse Sophie Germain