Qλ is the set of nonprincipal filters on ω which are generated by fewer than λ sets, for λ a fixed uncountable, regular cardinal ≤ c. We analyze forcing with Qλ, where Qλ is partially ordered in such a way that a filter F1 is more informative than F2 iff F1 includes F2. Qλ-forcing adjoins an ultrafilter on ω but adds no new reals. We analyze Qλ-forcing from a forcing-theoretic viewpoint. We also analyze the properties of Qλ-generic ultrafilters. These properties are independent of ZFC and depend very much on the ground model. In particular, we study Qλ-forcing over ground models which are Cohen real extensions, random real extensions, and models which satisfy Martin's Axiom.
In §2 we give notations and definitions, and review some of the basic facts about forcing and ultrafilters which we will use. In §3 we introduce Qλ-forcing and prove some basic lemmas about it. §4 studies Qc-forcing. §§5, 6, and 7 analyze Qλ-forcing over ground models of Martin's Axiom, ground models which are generated by Cohen reals, and ground models which are generated by random reals, respectively. Qλ-forcing over Cohen real and random real models is isomorphic to the notion of forcing which adjoins a Cohen generic subset of λ; this is proved in §8.