Measures that violate the Generalized Continuum Hypothesis

Abstract

A simple \(P_λ\)-point on a regular cardinal \(κ\) is a uniform ultrafilter on \(κ\) with a mod-bounded decreasing generating sequence of length \(λ\). We prove that if there is a simple $P_λ$-point ultrafilter over $κ>ω$, then $λ=\mathfrak{d}_κ=\mathfrak{b}_κ=\mathfrak{u}_κ=\mathfrak{r}_κ=\mathfrak{s}_κ$. 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 $P_{κ^{++}}$-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 $\mathfrak{t}_κ$ is greater than $κ^+$ and $κ$ 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 $2^κ>κ^+$.