Indecomposable ultrafilters and collapsing successors of singular cardinals

Seminar
Speaker
Inbar Oren (HUJI)
Date
06/02/2025 - 17:00 - 15:00Add to Calendar 2025-02-06 15:00:00 2025-02-06 17:00:00 Indecomposable ultrafilters and collapsing successors of singular cardinals In a famous paper by Ben David and Magidor, they show the existence of an (\aleph_0,\aleph_\omega)-indecomposable ultrafilter on \aleph_{\omega+1}, in an intermediate model of a forcing that collapses \aleph_{\omega+1}. This is an indication that an (\aleph_0,\aleph_\omega)-indecomposable ultrafilter on \aleph_{\omega+1} might be enough to collapse \aleph_{\omega+1} by a forcing extension. I will show that if \lambda is a singular cardinal, strong limit and there is an (\aleph_0,\lambda)-indecomposable ultrafilter on \lambda^+, then there is a \lambda^{++} c.c. forcing notion that collapses \lambda^+ and preserves \lambda. As a corollary we will get that there can't be a (\aleph_0,\lambda)-indecomposable ultrafilter on  \lambda^+ if cf(\lambda)>\omega_1, in partcular there is no  (\aleph_0,\aleph_{\omega_2})-indecomposable ultrafilter on \aleph_{\omega_2+1}. Seminar room אוניברסיטת בר-אילן - המחלקה למתמטיקה mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
Seminar room
Abstract

In a famous paper by Ben David and Magidor, they show the existence of an (\aleph_0,\aleph_\omega)-indecomposable ultrafilter on \aleph_{\omega+1}, in an intermediate model of a forcing that collapses \aleph_{\omega+1}. This is an indication that an (\aleph_0,\aleph_\omega)-indecomposable ultrafilter on \aleph_{\omega+1} might be enough to collapse \aleph_{\omega+1} by a forcing extension.
I will show that if \lambda is a singular cardinal, strong limit and there is an (\aleph_0,\lambda)-indecomposable ultrafilter on \lambda^+, then there is a \lambda^{++} c.c. forcing notion that collapses \lambda^+ and preserves \lambda.
As a corollary we will get that there can't be a (\aleph_0,\lambda)-indecomposable ultrafilter on  \lambda^+ if cf(\lambda)>\omega_1, in partcular there is no  (\aleph_0,\aleph_{\omega_2})-indecomposable ultrafilter on \aleph_{\omega_2+1}.

תאריך עדכון אחרון : 02/02/2025