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

Part of: Set theory

Published online by Cambridge University Press:  18 March 2025

Andreas Lietz*
Affiliation:
Universität Münster, Münster, Germany. 2023.
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Abstract

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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
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Footnotes

Supervised by Ralf Schindler.

References

Asperó, D. and Schindler, R., Martin’s Maximum ${}^{++}$ implies Woodin’s axiom $\left(\ast \right)$ . Annals of Mathematics (2) , vol. 193 (2021), pp. 793835.CrossRefGoogle Scholar
Usuba, T., Extendible cardinals and the mantle . Archive for Mathematical Logic , vol. 58 (2019), pp. 7175.CrossRefGoogle Scholar
Woodin, W., The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal , Walter de Gruyter, Berlin, 2010.CrossRefGoogle Scholar