A framework for forcing constructions at successors of singular cardinals

James Cummings, Mirna Dzamonja, Menachem Magidor, Charles Morgan, Saharon Shelah

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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 2^{κ^+} of graphs on κ^+ such that any graph on κ^+ embeds into one of the graphs in the collection.
Original languageEnglish
Pages (from-to)7405-7441
Number of pages37
JournalTransactions of the American Mathematical Society
Issue number10
Early online date31 May 2017
Publication statusPublished - Oct 2017


  • successor of singular cardinal
  • iterated forcing
  • strong chain condition
  • Radin forcing forcing axiom
  • universal graph
  • indestructible supercompat cardinal

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