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A framework for forcing constructions at successors of singular cardinals

Authors: James Cummings, Mirna Džamonja, Menachem Magidor, Charles Morgan and Saharon Shelah
Journal: Trans. Amer. Math. Soc. 369 (2017), 7405-7441
MSC (2010): Primary 03E35, 03E55, 03E75
Published electronically: May 31, 2017
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Abstract: We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal $ \kappa $ of uncountable cofinality, while $ \kappa ^+$ enjoys various combinatorial properties.

As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal $ \kappa $ of uncountable cofinality where SCH fails and such that there is a collection of size less than $ 2^{\kappa ^+}$ of graphs on $ \kappa ^+$ such that any graph on $ \kappa ^+$ embeds into one of the graphs in the collection.

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Additional Information

James Cummings
Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213

Mirna Džamonja
Affiliation: School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, United Kingdom

Menachem Magidor
Affiliation: Institute of Mathematics, Hebrew University of Jerusalem, 91904 Givat Ram, Israel

Charles Morgan
Affiliation: Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom — and — School of Mathematics, University of Edinburgh, Edinburgh, EH9 3FD, United Kingdom

Saharon Shelah
Affiliation: Institute of Mathematics, Hebrew University of Jerusalem, 91904 Givat Ram, Israel — and — Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08854

Keywords: Successor of singular cardinal, iterated forcing, strong chain condition, Radin forcing, forcing axiom, universal graph, indestructible supercompact cardinal
Received by editor(s): March 25, 2014
Received by editor(s) in revised form: October 9, 2015, April 1, 2016, and April 30, 2016
Published electronically: May 31, 2017
Additional Notes: The first author thanks the National Science Foundation for support through grant DMS-1101156. The first, second and fourth authors thank the Institut Henri Poincaré for support through the “Research in Paris” program during the period June 24–29, 2013. The second author thanks EPSRC for support through grants EP/G068720 and EP/I00498. The second, third and fifth authors thank the Mittag-Leffler Institute for support during the month of September 2009. The fourth author thanks EPSRC for support through grant EP/I00498. This publication is denoted [Sh963] in Saharon Shelah’s list of publications. Shelah thanks the United States-Israel Binational Science Foundation (grant no. 2006108), which partially supported this research.
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