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Diamond, square, and level by level equivalence

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Abstract.

We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with certain additional combinatorial properties. In particular, in this model, ♦ δ holds for every regular uncountable cardinal δ, and below the least supercompact cardinal κ, □ δ holds on a stationary subset of κ. There are no restrictions in our model on the structure of the class of supercompact cardinals.

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References

  1. Apter, A.: On Level by Level Equivalence and Inequivalence between Strong Compactness and Supercompactness. Fundamenta Mathematicae 171, 77–92 (2002)

    MATH  Google Scholar 

  2. Apter, A., Cummings, J.: Identity Crises and Strong Compactness II: Strong Cardinals. Arch. Math. Logic 40, 25–38 (2001)

    Article  MATH  Google Scholar 

  3. Apter, A., Shelah, S.: On the Strong Equality between Supercompactness and Strong Compactness. Trans. Am. Math. Soc. 349, 103–128 (1997)

    Article  MATH  Google Scholar 

  4. Cummings, J., Foreman, M., Magidor,M.: Squares, Scales, and Stationary Reflection. J. Math. Logic 1, 35–98 (2001)

    Article  MATH  Google Scholar 

  5. Hamkins, J.D.: Extensions with the Approximation and Cover Properties Have No New Large Cardinals. To appear in Fundamenta Mathematicae

  6. Hamkins, J.D.: Gap Forcing. Israel J. Math. 125, 237–252 (2001)

    MATH  Google Scholar 

  7. Hamkins, J.D.: Gap Forcing: Generalizing the Lévy-Solovay Theorem. Bulletin of Symbolic Logic 5, 264–272 (1999)

    MATH  Google Scholar 

  8. Jech, T.: Set Theory: The Third Millennium Edition, Revised and Expanded. Springer-Verlag, Berlin and New York, 2003

  9. Jensen, R.: The Fine Structure of the Constructible Hierarchy. Ann. Math. Logic 4, 229–308 (1972)

    Article  MATH  Google Scholar 

  10. Kanamori, A.: The Higher Infinite. Springer-Verlag, Berlin and New York, 1994

  11. Kimchi, Y., Magidor, M.: The Independence between the Concepts of Compactness and Supercompactness. Circulated manuscript

  12. Kunen, K.: Set Theory: An Introduction to Independence Proofs. Studies in Logic and the Foundations of Mathematics 102, North-Holland, Amsterdam and New York, 1980

  13. Lévy, A., Solovay, R.: Measurable Cardinals and the Continuum Hypothesis. Israel J. Math. 5, 234–248 (1967)

    Google Scholar 

  14. Magidor, M.: How Large is the First Strongly Compact Cardinal?. Ann. Math. Logic 10, 33–57 (1976)

    Article  MATH  Google Scholar 

  15. Menas, T.: On Strong Compactness and Supercompactness. Ann. Math. Logic 7, 327–359 (1974)

    Article  MATH  Google Scholar 

  16. Solovay, R.: Strongly Compact Cardinals and the GCH. in: Proceedings of the Tarski Symposium, Proceedings of Symposia in Pure Mathematics 25, American Mathematical Society, Providence, pp. 365–372 (1974)

  17. Solovay, R., Reinhardt, W., Kanamori, A.: Strong Axioms of Infinity and Elementary Embeddings. Ann. Math. Logic 13, 73–116 (1978)

    Article  MATH  Google Scholar 

Download references

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Authors and Affiliations

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

    Arthur W. Apter

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

Additional information

The author wishes to thank the referee for numerous helpful comments and suggestions, which have considerably improved the presentation of the material contained herein. The author also wishes to thank Andreas Blass, the corresponding editor, for a useful suggestion, and Grigor Sargsyan for a very helpful conversation on the subject matter of this paper.

Mathematics Subject Classification (2000): 03E35, 03E55

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Apter, A. Diamond, square, and level by level equivalence. Arch. Math. Logic 44, 387–395 (2005). https://doi.org/10.1007/s00153-004-0252-0

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  • DOI: https://doi.org/10.1007/s00153-004-0252-0

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