A framework for forcing constructions at successors of singular cardinals

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

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

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.

Original languageEnglish
Pages (from-to)7405-7411
Number of pages7
JournalTransactions of the American Mathematical Society
Volume369
Issue number10
DOIs
StatePublished - 2017

Bibliographical note

Publisher Copyright:
© 2017 American Mathematical Society.

Keywords

  • Forcing axiom
  • Indestructible supercompact cardinal
  • Iterated forcing
  • Radin forcing
  • Strong chain condition
  • Successor of singular cardinal
  • Universal graph

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