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Proceedings of the American Mathematical Society

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Singular cardinals and square properties


Authors: Menachem Magidor and Dima Sinapova
Journal: Proc. Amer. Math. Soc. 145 (2017), 4971-4980
MSC (2010): Primary 03E05, 03E35, 03E55
DOI: https://doi.org/10.1090/proc/13650
Published electronically: June 5, 2017
MathSciNet review: 3692010
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Abstract: We analyze the effect of singularizing cardinals on square properties. By work of Džamonja-Shelah and of Gitik, if you singularize an inaccessible cardinal to countable cofinality while preserving its successor, then $\square _{\kappa , \omega }$ holds in the bigger model. We extend this to the situation where every regular cardinal in an interval $[\kappa ,\nu ]$ is singularized, for some regular cardinal $\nu$. More precisely, we show that if $V\subset W$, $\kappa <\nu$ are cardinals, where $\nu$ is regular in $V$, $\kappa$ is a singular cardinal in $W$ of countable cofinality, $\mathrm {cf}^W(\tau )=\omega$ for all $V$-regular $\kappa \leq \tau \leq \nu$, and $(\nu ^+)^V=(\kappa ^+)^W$, then $W\models \square _{\kappa ,\omega }$.


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Additional Information

Menachem Magidor
Affiliation: Department of Mathematics, University of the Negev, Be’er Sheva, Israel
MR Author ID: 118010
ORCID: 0000-0002-5568-8397

Dima Sinapova
Affiliation: Department of Mathematics, University of Illinois at Chicago, Chicago, Illinois 60607
MR Author ID: 813838

Received by editor(s): October 8, 2014
Received by editor(s) in revised form: October 20, 2015, December 28, 2015, and December 10, 2016
Published electronically: June 5, 2017
Additional Notes: The second author was partially supported by the National Science Foundation under Grant No. DMS - $1362485$
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2017 American Mathematical Society