Measures that violate the Generalized Continuum Hypothesis

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 $2^{\kappa }>{\kappa }^{+}$.