Strongly proper forcing and some problems of Foreman

TL;DR

This paper addresses several open problems posed by Foreman related to ideals and strongly proper forcing, providing new solutions and clarifications that advance understanding in set theory and forcing techniques.

Contribution

It offers new solutions to Foreman's problems on ideals, projective antichain catching, and the relationship between generic hugeness and almost hugeness, using strongly proper forcing.

Findings

Presaturation implies projective antichain catching.

Solved the relationship between generic hugeness and almost hugeness.

Provided solutions to technical questions on Foreman's Duality Theorem.

Abstract

We provide solutions to several problems of Foreman about ideals, several of which are closely related to Mitchell's notion of \emph{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~\cite{MR2768692} about ideal projections which is more comprehensive and simpler than the solution obtained in \cite{MR3343538}. 2) We solve an older question from Foreman~\cite{MR819932} about the relationship between generic hugeness and generic almost hugeness. 3) Finally, we provide solutions to two technical questions from Foreman~\cite{MR3038554} and \cite{MR2768692} related to his \emph{Duality Theorem}.