A Framework for Forcing Constructions at Successors of Singular Cardinals

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

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arXiv · 1403.6795

A Framework for Forcing Constructions at Successors of Singular Cardinals

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

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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 of uncountable cofinality, while its successor 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 $\kappa$ of uncountable cofinality where SCH fails and for which there is a collection of graphs on $\kappa^+$ whose size is less than $2^\kappa$ and such that any graph on $\kappa^+$ embeds into one of the graphs in the collection.

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BibTeX RIS

James Cummings, Mirna Džamonja, Menachem Magidor, Charles Morgan, Saharon Shelah. 2014-03-26. A Framework for Forcing Constructions at Successors of Singular Cardinals. https://arxiv.org/abs/1403.6795

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