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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
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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

Dima Sinapova
Affiliation: Department of Mathematics, University of Illinois at Chicago, Chicago, Illinois 60607

DOI: https://doi.org/10.1090/proc/13650
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

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