Destroying Saturation while Preserving Presaturation at an Inaccessible; an Iterated Forcing Argument

TL;DR

This paper demonstrates a method to de-saturate presaturated ideals at inaccessible cardinals while maintaining presaturation, using an iterated forcing technique and Foreman's Duality Theorem.

Contribution

It introduces a novel iterated forcing approach to manipulate presaturated ideals at inaccessible cardinals, answering longstanding open questions.

Findings

Successfully de-saturated a large class of presaturated ideals

Preservation of presaturation during de-saturation process

Application of Foreman's Duality Theorem in forcing arguments

Abstract

We prove that a large class of presaturated ideals at inaccessible cardinals can be de-saturated while preserving their presaturation, answering both a question of Foreman and of Cox and Eskew. We do so by iterating a generalized version of Baumgartner and Taylor's forcing to add a club with finite conditions along an inaccessible cardinal, and invoking Foreman's Duality Theorem.