Strongly proper forcing and some problems of Foreman

Abstract

We answer several questions of Foreman, most of which are closely related to Mitchell’s notion of strongly proper forcing. We prove that presaturation of a normal ideal implies projective antichain catching, providing a solution to a problem of Foreman about ideal projections that is more comprehensive and simpler than the earlier solution obtained by Cox and Zeman. We answer an older question of Foreman about the relationship between generic hugeness and generic almost hugeness. Finally, we answer two technical questions of Foreman related to his Duality Theorem.