Strongly proper forcing and some problems of Foreman
Abstract
We provide solutions to several problems of Foreman about ideals, several of which are closely related to Mitchell's notion of strongly proper forcing. We prove: 1) Presaturation of a normal ideal implies projective antichain catching, enabling us to provide a solution to a problem from Foreman~MR2768692 about ideal projections which is more comprehensive and simpler than the solution obtained in MR3343538. 2) We solve an older question from Foreman~MR819932 about the relationship between generic hugeness and generic almost hugeness. 3) Finally, we provide solutions to two technical questions from Foreman~MR3038554 and MR2768692 related to his Duality Theorem.
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