Measures that violate the Generalized Continuum Hypothesis

Abstract

A simple \(P_\lambda\)-point on a regular cardinal \(\kappa\) is a uniform ultrafilter on \(\kappa\) with a mod-bounded decreasing generating sequence of length \(\lambda\). We prove that if there is a simple $P_\lambda$-point ultrafilter over $\kappa>\omega$, then $\lambda=\mathfrak{d}_\kappa=\mathfrak{b}_\kappa=\mathfrak{u}_\kappa=\mathfrak{r}_\kappa=\mathfrak{s}_\kappa$. We show that such ultrafilters appear in the models of \cite{SimonOmer,BROOKETAYLOR201737}. We improve the lower bound for the consistency strength of the existence of a $P_{\kappa^{++}}$-point to a $2$-strong cardinal. Finally, we apply our arguments to obtain non-trivial lower bounds for (1) the statement that the generalized tower number $\mathfrak{t}_\kappa$ is greater than $\kappa^+$ and $\kappa$ is measurable, (2) the preservation of measurability after the generalized Mathias forcing, and (3) variations of filter games of \cite{NIELSEN_WELCH_2019,HolySchlicht:HierarchyRamseyLikeCardinals,MagForZem} in the case $2^\kappa>\kappa^+$.