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On some questions concerning strong compactness

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Abstract

A question of Woodin asks if κ is strongly compact and GCH holds below κ, then must GCH hold everywhere? One variant of this question asks if κ is strongly compact and GCH fails at every regular cardinal δ < κ, then must GCH fail at some regular cardinal δ ≥ κ? Another variant asks if it is possible for GCH to fail at every limit cardinal less than or equal to a strongly compact cardinal κ. We get a negative answer to the first of these questions and positive answers to the second of these questions for a supercompact cardinal κ in the context of the absence of the full Axiom of Choice.

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References

  1. Apter A.: A note on strong compactness and supercompactness. Bull. Lond. Math. Soc. 23, 113–115 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  2. Apter A.: On a problem of Woodin. Arch. Math. Log. 39, 253–259 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bull E.: Successive large cardinals. Ann. Math. Log. 15, 161–191 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  4. Gitik, M.: Prikry-type forcings. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set Theory, pp. 1351–1448. Springer, Berlin (2010)

  5. Jech T.: Set Theory. The Third Millennium Edition, Revised and Expanded. Springer, Berlin (2003)

    MATH  Google Scholar 

  6. Kanamori A.: The Higher Infinite. Springer, Berlin (1994)

    MATH  Google Scholar 

  7. Laver R.: Making the supercompactness of κ indestructible under κ-directed closed forcing. Isr. J. Math. 29, 385–388 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lévy A., Solovay R.: Measurable cardinals and the continuum hypothesis. Isr. J. Math. 5, 234–248 (1967)

    Article  MATH  Google Scholar 

  9. Radin L.: Adding closed cofinal sequences to large cardinals. Ann. Math. Log. 22, 243–261 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  10. Solovay, R.: Strongly compact cardinals and the GCH. In: Proceedings of the Tarski Symposium, Proceedings of Symposia in Pure Mathematics, vol. 25, pp. 365–372. American Mathematical Society, Providence (1974)

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Authors and Affiliations

  1. Department of Mathematics, Baruch College of CUNY, New York, NY, 10010, USA

    Arthur W. Apter

  2. The CUNY Graduate Center, Mathematics, 365 Fifth Avenue, New York, NY, 10016, USA

    Arthur W. Apter

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  1. Arthur W. Apter
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Correspondence to Arthur W. Apter.

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The author’s research was partially supported by PSC-CUNY grants.

The author wishes to thank Brent Cody for helpful conversations on the subject matter of this paper.

The author also wishes to thank the second referee for helpful corrections and suggestions which were incorporated into the current version of the paper.

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Apter, A.W. On some questions concerning strong compactness. Arch. Math. Logic 51, 819–829 (2012). https://doi.org/10.1007/s00153-012-0300-0

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  • DOI: https://doi.org/10.1007/s00153-012-0300-0

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