Filters and large cardinals Original Research Article

Assuming the consistency of the theory “ZFC + there exists a measurable cardinal”, we construct 1. (1) a model in which the first cardinal κ, such that 2κ > κ+, bears a normal filter F whose associated boolean algebra is κ+-distributive (and indeed strongly κ+-distributive as defined in Section 5), 2. (2) a model where there is a measurable cardinal κ such that, for every regular cardinal ρ < κ, 2ρ = ρ++ holds, 3. (3) a model of “ZFC + GCH” where there exists a non-measurable cardinal κ bearing a normal filter F whose associated boolean algebra is κ+-distributive (and κ+-saturated as well).