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On the consistency strength of level by level inequivalence

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Abstract

We show that the theories “ZFC \(+\) There is a supercompact cardinal” and “ZFC \(+\) There is a supercompact cardinal \(+\) Level by level inequivalence between strong compactness and supercompactness holds” are equiconsistent.

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References

  1. Apter, A.: Indestructible strong compactness and level by level inequivalence. Math. Log. Q. 59, 371–377 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Apter, A.: Level by level inequivalence beyond measurability. Arch. Math. Log. 50, 707–712 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Apter, A.: Level by level inequivalence, strong compactness, and GCH. Bull. Pol. Acad. Sci. Math. 60, 201–209 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Apter, A.: On level by level equivalence and inequivalence between strong compactness and supercompactness. Fundam. Math. 171, 77–92 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Apter, A.: Tallness and level by level equivalence and inequivalence. Math. Log. Q. 56, 4–12 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Apter, A., Gitman, V., Hamkins, J.D.: Inner models with large cardinal features usually obtained by forcing. Arch. Math. Log. 51, 257–283 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Apter, A., Hamkins, J.D.: Exactly controlling the non-supercompact strongly compact cardinals. J. Symb. Log. 68, 669–688 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Apter, A., Shelah, S.: On the strong equality between supercompactness and strong compactness. Trans. Am. Math. Soc. 349, 103–128 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cummings, J.: A model in which GCH holds at successors but fails at limits. Trans. Am. Math. Soc. 329, 1–39 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Di Prisco, C., Henle, J.: On the compactness of \(\aleph _1\) and \(\aleph _2\). J. Symb. Log. 43, 394–401 (1978)

    Article  Google Scholar 

  11. Foreman, M.: More saturated ideals. Cabal seminar 79–81. Lecture Notes in Mathematics 1019, pp. 1–27. Springer, Berlin (1983)

  12. Hamkins, J.D.: Gap forcing. Isr. J. Math. 125, 237–252 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hamkins, J.D.: Gap forcing: generalizing the Lévy–Solovay theorem. Bull. Symb. Log. 5, 264–272 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kanamori, A.: The Higher Infinite. Springer, Berlin (2009, 2003, 1994)

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

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

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

    Arthur W. Apter

  2. Department of Mathematics, The CUNY Graduate Center, 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.

Additional information

This paper is dedicated to the memory of Jim Baumgartner, a friend and inspiration to all those who knew him.

The author’s research was partially supported by PSC-CUNY Grants. The author wishes to thank Norman Perlmutter for a helpful conversation on the subject matter of this paper.

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Apter, A.W. On the consistency strength of level by level inequivalence. Arch. Math. Logic 56, 715–723 (2017). https://doi.org/10.1007/s00153-017-0538-7

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  • DOI: https://doi.org/10.1007/s00153-017-0538-7

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