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Ordinal definable subsets of singular cardinals

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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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Authors and Affiliations

  1. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213-3890, USA

    James Cummings

  2. Kurt Gödel Research Center for Mathematical Logic, University of Vienna, A-1090, Vienna, Austria

    Sy-David Friedman

  3. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel

    Menachem Magidor

  4. Department of Mathematics, Bar-Ilan University, Ramat-Gan, 5290002, Israel

    Assaf Rinot

  5. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607-7045, USA

    Dima Sinapova

Authors
  1. James Cummings
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  2. Sy-David Friedman
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  3. Menachem Magidor
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  4. Assaf Rinot
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  5. Dima Sinapova
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Corresponding author

Correspondence to James Cummings.

Additional information

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.

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Cummings, J., Friedman, SD., Magidor, M. et al. Ordinal definable subsets of singular cardinals. Isr. J. Math. 226, 781–804 (2018). https://doi.org/10.1007/s11856-018-1712-2

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  • DOI: https://doi.org/10.1007/s11856-018-1712-2

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