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Forcing “
$\mathrm {NS}_{\omega _1}$ is
$\omega _1$-Dense” from Large Cardinals
Published online by Cambridge University Press: 18 March 2025
Abstract
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 Lietz
E-mail: [email protected].
URL: https://andreas-lietz.github.io/resources/PDFs/AJourneyGuidedByThe Stars.pdf.
- Type
- Thesis Abstract
- Information
- Copyright
- © The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
Footnotes
Supervised by Ralf Schindler.
References

