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ON COMPACTNESS OF WEAK SQUARE AT SINGULARS OF UNCOUNTABLE COFINALITY

Part of: Set theory

Published online by Cambridge University Press:  04 January 2024

MAXWELL LEVINE
MAXWELL LEVINE*
Affiliation:
INSTITUTE OF MATHEMATICS UNIVERSITY OF FREIBURG ERNST-ZERMELO-STRASSE 1, 79104 FREIBURG IM BREISGAU, GERMANY
*
E-mail: maxwell.levine@mathematik.uni-freiburg.de
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Abstract

Cummings, Foreman, and Magidor proved that Jensen’s square principle is non-compact at $\aleph _\omega $, meaning that it is consistent that $\square _{\aleph _n}$ holds for all $n<\omega $ while $\square _{\aleph _\omega }$ fails. We investigate the natural question of whether this phenomenon generalizes to singulars of uncountable cofinality. Surprisingly, we show that under some mild ${{\mathsf {PCF}}}$-theoretic hypotheses, the weak square principle $\square _\kappa ^*$ is in fact compact at singulars of uncountable cofinality.

Keywords

PCF theorylarge cardinalsindependence proofs

MSC classification

Primary: 03E04: Ordered sets and their cofinalities; pcf theory
Secondary: 03E35: Consistency and independence results 03E55: Large cardinals
Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 11
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

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