Measures that violate the generalized continuum hypothesis

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 = d κ = b κ = u κ = r κ = s κ . We show that such ultrafilters appear in the models of [3] , [13] . 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 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 [28] , [22] , [18] in the case 2 κ > κ + .