arXiv:1403.6795v1 [math.LO] 26 Mar 2014

arXiv:1403.6795v1 [math.LO] 26 Mar 2014

A FRAMEWORK FOR FORCING CONSTRUCTIONS AT SUCCESSORS OF SINGULAR CARDINALS JAMES CUMMINGS, MIRNA DŽAMONJA, MENACHEM MAGIDOR, CHARLES MORGAN, AND SAHARON SHELAH

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 κ of uncountable cofinality, while κ+ 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 κ of uncountable cofinality where SCH fails + and such that there is a collection of size less than 2κ of graphs on κ+ such that any graph on κ+ embeds into one of the graphs in the collection.

Introduction The class of uncountable regular cardinals is naturally divided into three disjoint classes: the successors of regular cardinals, the successors of singular cardinals and the weakly inaccessible cardinals. When we 2010 Mathematics Subject Classification. Primary: 03E35, 03E55, 03E75. Key words and phrases. successor of singular, Radin forcing, forcing axiom, universal graph. Cummings thanks the National Science Foundation for their support through grant DMS-1101156. Cummings, Džamonja and Morgan thank the Institut Henri Poincaré for their support through the “Research in Paris” program during the period 24-29 June 2013. Džamonja, Magidor and Shelah thank the Mittag-Leffler Institute for their support during the month of September 2009. Mirna Džamonja thanks EPSRC for their support through their grants EP/G068720 and EP/I00498. Morgan thanks EPSRC for their 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. We thank Jacob Davis for his attentive reading of and comments on this paper. 1

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CUMMINGS, DŽAMONJA, MAGIDOR, MORGAN, AND SHELAH

consider a combinatorial question about uncountable regular cardinals, typically these classes require separate treatment and very frequently the successors of singular cardinals present the hardest problems. In particular there are subtle constraints (for example in cardinal arithmetic) on the combinatorics of successors of singular cardinals, and consistency results in this area often involve large cardinals. To give some context for our work, we review a standard strategy for proving consistency results about the successors of regular cardinals. This strategy involves iterating