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A framework for forcing constructions at successors of singular cardinals

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journal contribution
posted on 01.03.2014, 00:00 by James CummingsJames Cummings, Mirna Dzamonja, Menachim Magidor, Charles Morgan, Saharon Shelah

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.

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01/03/2014

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