Ordinal definable subsets of singular cardinals |
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| Authors: | James Cummings Sy-David Friedman Menachem Magidor Assaf Rinot Dima Sinapova |
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| Institution: | 1.Department of Mathematical Sciences,Carnegie Mellon University,Pittsburgh,USA;2.Kurt G?del Research Center for Mathematical Logic,University of Vienna,Vienna,Austria;3.Einstein Institute of Mathematics,The Hebrew University of Jerusalem,Jerusalem,Israel;4.Department of Mathematics,Bar-Ilan University,Ramat-Gan,Israel;5.Department of Mathematics, Statistics, and Computer Science,University of Illinois at Chicago,Chicago,USA |
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| Abstract: | A remarkable result by Shelah states that if κ is a singular strong limit cardinal of uncountable cofinality, then there is a subset x of κ such that HODx contains the power set of κ. We develop a version of diagonal extender-based supercompact Prikry forcing, and use it to show that singular cardinals of countable cofinality do not in general have this property, and in fact it is consistent that for some singular strong limit cardinal κ of countable cofinality κ+ is supercompact in HODx for all x ? κ. |
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