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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Singular cardinals and square properties
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by Menachem Magidor and Dima Sinapova PDF
Proc. Amer. Math. Soc. 145 (2017), 4971-4980 Request permission

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
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 4971-4980
  • MSC (2010): Primary 03E05, 03E35, 03E55
  • DOI: https://doi.org/10.1090/proc/13650
  • MathSciNet review: 3692010