Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

Request Permissions   Purchase Content 
 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 }$.


References [Enhancements On Off] (What's this?)

  • [1] James Cummings, Matthew Foreman, and Menachem Magidor, Squares, scales and stationary reflection, J. Math. Log. 1 (2001), no. 1, 35-98. MR 1838355, https://doi.org/10.1142/S021906130100003X
  • [2] James Cummings and Ernest Schimmerling, Indexed squares, Israel J. Math. 131 (2002), 61-99. MR 1942302, https://doi.org/10.1007/BF02785851
  • [3] Mirna Džamonja and Saharon Shelah, On squares, outside guessing of clubs and $ I_{<f}[\lambda ]$, Fund. Math. 148 (1995), no. 2, 165-198. MR 1360144
  • [4] Moti Gitik, Some results on the nonstationary ideal. II, Israel J. Math. 99 (1997), 175-188. MR 1469092, https://doi.org/10.1007/BF02760681
  • [5] Moti Gitik and Assaf Sharon, On SCH and the approachability property, Proc. Amer. Math. Soc. 136 (2008), no. 1, 311-320. MR 2350418, https://doi.org/10.1090/S0002-9939-07-08716-3
  • [6] Saharon Shelah, Cardinal arithmetic, Oxford Logic Guides, vol. 29, The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR 1318912

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 03E05, 03E35, 03E55

Retrieve articles in all journals with MSC (2010): 03E05, 03E35, 03E55


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

American Mathematical Society