Complete ineffability is a classical topic in Set Theory, much of it due to Baumgartner. A cardinal kappa being completely ineffable implies that kappa is inaccessible.  Work of Eshkol, Foreman and Magidor has shown that ineffability is equivalent to the existence of certain threading ideals. 

This talk describes a new kind of ideal: weakly threading ideals, and how to show that it is consistent that they exist on successor cardinals such as aleph_2 assuming the consistency of a measurable cardinal.