Yair Hayut: Filter compactness and squares

BIU seminar in Set Theory

April 8, 2018

Speaker: Yair Hayut (TAU)

Title: Filter compactness and squares

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.

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