Ordinal definable subsets of singular cardinals

James Cummings; Sy David Friedman; Menachem Magidor; Assaf Rinot; Dima Sinapova

doi:10.1007/s11856-018-1712-2

Ordinal definable subsets of singular cardinals

James Cummings, Sy David Friedman, Menachem Magidor, Assaf Rinot, Dima Sinapova

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

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 HOD_x 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 HOD_x for all x ⊆ κ.

Original language	English
Pages (from-to)	781-804
Number of pages	24
Journal	Israel Journal of Mathematics
Volume	226
Issue number	2
DOIs	https://doi.org/10.1007/s11856-018-1712-2
State	Published - 1 Jun 2018

Bibliographical note

Publisher Copyright:
© 2018, Hebrew University of Jerusalem.

Funding

∗Cummings was partially supported by the National Science Foundation, grants DMS-1101156 and DMS-1500790. ∗∗ Friedman would like to thank the Austrian Science Fund for its generous support through the research project P25748. † Rinot was partially supported by the Israel Science Foundation, grant 1630/14. ††Sinapova was partially supported by the National Science Foundation, grants DMS-1362485 and Career-1454945. Received April 15, 2016 and in revised form July 25, 2017

Funders	Funder number
National Science Foundation	DMS-1101156, DMS-1500790
Austrian Science Fund	P25748
Israel Science Foundation	DMS-1362485, Career-1454945, 1630/14

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10.1007/s11856-018-1712-2

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Ordinal definable subsets of singular cardinals

Abstract

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