A framework for forcing constructions at successors of singular cardinals

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

    Research output: Contribution to journalArticlepeer-review

    6 Citations (Scopus)
    17 Downloads (Pure)

    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 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
    Volume369
    Issue number10
    Early online date31 May 2017
    DOIs
    Publication statusPublished - Oct 2017

    Keywords

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

    Cite this