Filter compactness and squares

Seminar
Speaker
Yair Hayut (TAU)
Date
09/04/2018 - 15:00 - 13:00Add to Calendar 2018-04-09 13:00:00 2018-04-09 15:00:00 Filter compactness and squares Strongly compact cardinals are characterized by the property that  any $\kappa$-complete filter can be extended to a $\kappa$-complete ultrafilter. When restricting the cardinality of the underlying set, we obtain a nontrivial hierarchy. For example, when requiring the extension property to hold only for filters on $\kappa$, we obtain Gitik's $\kappa$-compact cardinals, which are known to be consistently weaker than $\kappa$ being $\kappa^+$-strongly compact. In this talk I will focus on the level by level connection between the filter extension property and the compactness for $L_{\kappa,\kappa}$. Using the compactness, I will show that if $\kappa$ is $\kappa$-compact then $\square(\kappa^{+})$-fails. Building 507, Room 204 אוניברסיטת בר-אילן - Department of Mathematics mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
Building 507, Room 204
Abstract

Strongly compact cardinals are characterized by the property that  any $\kappa$-complete filter can be extended to a $\kappa$-complete ultrafilter. When restricting the cardinality of the underlying set, we obtain a nontrivial hierarchy. For example, when requiring the extension property to hold only for filters on $\kappa$, we obtain Gitik's $\kappa$-compact cardinals, which are known to be consistently weaker than $\kappa$ being $\kappa^+$-strongly compact.

In this talk I will focus on the level by level connection between the filter extension property and the compactness for $L_{\kappa,\kappa}$. Using the compactness, I will show that if $\kappa$ is $\kappa$-compact then $\square(\kappa^{+})$-fails.

Last Updated Date : 22/03/2018