Journal Article

Forcing “\\(\\mathrm {NS}_{\\omega _1}\\) is \\(\\omega _1\\)-Dense” from Large Cardinals

Lietz, Andreas

2024

Overview

We answer a question of Woodin [3] by showing that “\\(\\mathrm {NS}_{\\omega _1}\\) is \\(\\omega _1\\)-dense” holds in a stationary set preserving extension of any universe with a cardinal \\(\\kappa \\) which is a limit of \\({<}\\kappa \\)-supercompact cardinals. We introduce a new forcing axiom \\(\\mathrm {Q}\\)-Maximum, prove it consistent from a supercompact limit of supercompact cardinals, and show that it implies the version of Woodin’s \\((*)\\)-axiom for \\(\\mathbb Q_{\\mathrm {max}}\\). It follows that \\(\\mathrm {Q}\\)-Maximum implies “\\(\\mathrm {NS}_{\\omega _1}\\) is \\(\\omega _1\\)-dense.” Along the way we produce a number of other new instances of Asperó–Schindler’s \\(\\mathrm {MM}^{++}\\Rightarrow (*)\\) (see [1]).To force \\(\\mathrm {Q}\\)-Maximum, we develop a method which allows for iterating \\(\\omega _1\\)-preserving forcings which may destroy stationary sets, without collapsing \\(\\omega _1\\). We isolate a new regularity property for \\(\\omega _1\\)-preserving forcings called respectfulness which lies at the heart of the resulting iteration theorem.In the second part, we show that the \\(\\kappa \\)-mantle, i.e., the intersection of all grounds which extend to V via forcing of size \\({<}\\kappa \\), may fail to be a model of \\(\\mathrm {AC}\\) for various types of \\(\\kappa \\). Most importantly, it can be arranged that \\(\\kappa \\) is a Mahlo cardinal. This answers a question of Usuba [2].Abstract prepared by Andreas LietzE-mail: [e-mail]: https://andreas-lietz.github.io/resources/PDFs/AJourneyGuidedByThe Stars.pdf.

• Publisher: Cambridge University Press
• Subject: Questions