Science China Mathematics

, Volume 61, Issue 4, pp 677–684 | Cite as

Distributive proper forcing axiom and a left-right dichotomy of Cichoń’s diagram

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Abstract

In this paper, we study distributive proper forcing axiom (DPFA) and prove its consistency with a dichotomy of the Cichoń’s diagram, relative to certain large cardinal assumption. Namely, we evaluate the cardinal invariants in Cichoń’s diagram with the first two uncountable cardinals in the way that the left-hand side has the least possible cardinality while the right-hand side has the largest possible value, and preserve the evaluation along the way of forcing DPFA.

Keywords

forcing axiom Cichoń’s diagram preservation 

MSC(2010)

03E17 03E35 03E50 

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Notes

Acknowledgements

The author thanks Institute of Mathematical Science, National University of Singapore for support. The author thanks the reviewers for the suggestions and comments which greatly improved the manuscript. The author is inspired by many people when writing this paper. Among others, the author thanks Professor Qi Feng for his consistent encouragement. Last but foremost, the author thanks his parents for their love and dedicates this work to them.

References

  1. 1.
    Bartoszynski T, Judah H. Set Theory. Wellesley: A K Peters, 1995MATHGoogle Scholar
  2. 2.
    Baumgartner J. Applications of the proper forcing axiom. In: Handbook of Set-Theoretic Topology. Amsterdam: North-Holland, 1984, 913–959CrossRefGoogle Scholar
  3. 3.
    Baumgartner J, Harrington L, Kleinberg E. Adding a closed unbounded set. Fund Math, 1973, 79: 101–106MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Blass A. Combinatorial Cardinal Characteristics of the Continuum. Handbook of Set Theory. New York: Springer, 2010MATHGoogle Scholar
  5. 5.
    Cohen P. The independence of the continuum hypothesis. I. Proc Natl Acad Sci USA, 1963, 50: 1143–1148MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Goldstern M. Tools for your forcing construction. In: Israel Mathematical Conference Proceedings. Set Theory of the Reals, vol. 6. Providence: Amer Math Soc, 1993, 305–360MathSciNetMATHGoogle Scholar
  7. 7.
    Jech T. Set Theory, 3rd ed. Springer Monographs in Mathematics. New York: Springer-Verlag, 2003MATHGoogle Scholar
  8. 8.
    Kunen K. Set Theory: An Introduction to Independence Proofs. Studies in Logic and the Foundations of Mathematics, vol. 102. Amsterdam: North-Holland, 1980Google Scholar
  9. 9.
    Laver R. Making the supercompactness of k indestructible under k-directed closed forcing. Israel J Math, 1978, 29: 385–388MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Martin D, Solovay R. Internal Cohen extensions. Ann Math Logic, 1970, 2: 143–178MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Moore J. Set mapping re ection. J Math Log, 2005, 51: 89–97CrossRefGoogle Scholar
  12. 12.
    Shelah S. Proper Forcing. Lecture Notes in Mathematics, vol. 940. New York: Springer-Verlag, 1982Google Scholar
  13. 13.
    Todorcevic S. Partition Problems in Topology. Contemporary Mathematics, vol. 84. Providence: Amer Math Soc, 1989Google Scholar
  14. 14.
    Zhu H. Distributive proper forcing axiom. PhD Thesis. Singapore: National University of Singapore, 2012Google Scholar
  15. 15.
    Zhu H. Distributive proper forcing axiom and cardinal invariants. Arch Math Logic, 2013, 52: 497–506MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Data and Computer ScienceSun Yat-sen UniversityGuangzhouChina

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