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Large cardinals with few measures


Authors: Arthur W. Apter, James Cummings and Joel David Hamkins
Journal: Proc. Amer. Math. Soc. 135 (2007), 2291-2300
MSC (2000): Primary 03E35, 03E55
DOI: https://doi.org/10.1090/S0002-9939-07-08786-2
Published electronically: March 2, 2007
MathSciNet review: 2299507
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Abstract | References | Similar Articles | Additional Information

Abstract: We show, assuming the consistency of one measurable cardinal, that it is consistent for there to be exactly $ \kappa^+$ many normal measures on the least measurable cardinal $ \kappa$. This answers a question of Stewart Baldwin. The methods generalize to higher cardinals, showing that the number of $ \lambda$ strong compactness or $ \lambda$ supercompactness measures on $ P_\kappa(\lambda)$ can be exactly $ \lambda^+$ if $ \lambda > \kappa$ is a regular cardinal. We conclude with a list of open questions. Our proofs use a critical observation due to James Cummings.


References [Enhancements On Off] (What's this?)

  • 1. A. Apter, ``Some Remarks on Normal Measures and Measurable Cardinals'', Mathematical Logic Quarterly 47, 2001, 35-44. MR 1808944 (2001k:03105)
  • 2. A. Apter, J. Cummings, ``Identity Crises and Strong Compactness'', Journal of Symbolic Logic 65, 2000, 1895-1910. MR 1812190 (2001k:03104)
  • 3. S. Baldwin, ``The $ \triangleleft$-Ordering on Normal Ultrafilters'', Journal of Symbolic Logic 51, 1985, 936-952. MR 0820124 (87d:03124)
  • 4. J. Cummings, ``Possible Behaviours for the Mitchell Ordering'', Annals of Pure and Applied Logic 65, 1993, 107-123. MR 1257466 (95e:03146)
  • 5. J. D. Hamkins, ``A Class of Strong Diamond Principles'', arXiv:math.LO/0211419.
  • 6. J. D. Hamkins, ``Extensions with the Approximation and Cover Properties Have No New Large Cardinals'', Fundamenta Mathematicae 180, 2003, 257-277. MR 2063629 (2005m:03100)
  • 7. J. D. Hamkins, ``Gap Forcing'', Israel Journal of Mathematics 125, 2001, 237-252. MR 1853813 (2002h:03111)
  • 8. J. D. Hamkins, ``Small Forcing Makes any Cardinal Superdestructible'', Journal of Symbolic Logic 63, 1998, 51-58. MR 1607499 (99b:03068)
  • 9. J. D. Hamkins, ``The Lottery Preparation'', Annals of Pure and Applied Logic 101, 2000, 103-146.
  • 10. J. D. Hamkins, S. Shelah, ``Superdestructibility: A Dual to Laver's Indestructibility'', Journal of Symbolic Logic 63, 1998, 549-554. MR 1625927 (99m:03106)
  • 11. T. Jech, Set Theory: The Third Millennium Edition, Revised and Expanded, Springer-Verlag, Berlin and New York, 2003. MR 1940513 (2004g:03071)
  • 12. K. Kunen, ``Some Applications of Iterated Ultrapowers in Set Theory'', Annals of Mathematical Logic 1, 1970, 179-227. MR 0277346 (43:3080)
  • 13. K. Kunen, J. Paris, ``Boolean Extensions and Measurable Cardinals'', Annals of Mathematical Logic 2, 1971, 359-377. MR 0277381 (43:3114)
  • 14. A. Lévy, R. Solovay, ``Measurable Cardinals and the Continuum Hypothesis'', Israel Journal of Mathematics 5, 1967, 234-248. MR 0224458 (37:57)
  • 15. M. Magidor, ``How Large is the First Strongly Compact Cardinal?'', Annals of Mathematical Logic 10, 1976, 33-57. MR 0429566 (55:2578)
  • 16. W. Mitchell, ``Sets Constructible from Sequences of Ultrafilters'', Journal of Symbolic Logic 39, 1974, 57-66. MR 0344123 (49:8863)

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Additional Information

Arthur W. Apter
Affiliation: Department of Mathematics, Baruch College of CUNY, New York, New York 10010
Email: awapter@alum.mit.edu

James Cummings
Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Email: jcumming@andrew.cmu.edu

Joel David Hamkins
Affiliation: Mathematics Program, The Graduate Center of The City University of New York, 365 Fifth Avenue, New York, New York 10016 — Department of Mathematics, The College of Staten Island of CUNY, Staten Island, New York 10314
Email: jdh@hamkins.org

DOI: https://doi.org/10.1090/S0002-9939-07-08786-2
Keywords: Supercompact cardinal, strongly compact cardinal, measurable cardinal, normal measure.
Received by editor(s): March 14, 2006
Published electronically: March 2, 2007
Additional Notes: The research of the first and third authors was partially supported by PSC-CUNY grants and CUNY Collaborative Incentive Grants. The second author’s research was partially supported by NSF Grant DMS-0400982.
Communicated by: Julia Knight
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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