NAMBA FORCING, WEAK APPROXIMATION, AND GUESSING.

We prove a variation of Easton's lemma for strongly proper forcings, and use it to prove that, unlike the stronger principle IGMP, GMP together with 2 ^ω ≤ ω ₂ is consistent with the existence of an ω ₁-distributive nowhere c.c.c. forcing poset of size ω ₁. We introduce the idea of a weakly guessing model, and prove that many of the strong consequences of the principle GMP follow from the existence of stationarily many weakly guessing models. Using Namba forcing, we construct a model in which there are stationarily many indestructibly weakly guessing models which have a bounded countable subset not covered by any countable set in the model. [ABSTRACT FROM AUTHOR]