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Matrix iterations and Cichon’s diagram

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Abstract

Using matrix iterations of ccc posets, we prove the consistency with ZFC of some cases where the cardinals on the right hand side of Cichon’s diagram take two or three arbitrary values (two regular values, the third one with uncountable cofinality). Also, mixing this with the techniques in J Symb Log 56(3):795–810, 1991, we can prove that it is consistent with ZFC to assign, at the same time, several arbitrary regular values on the left hand side of Cichon’s diagram.

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References

  1. Bartoszynski T., Judah H.: Set Theory. On the Structure of the Real Line. A. K. Peters, Massachusetts (1995)

    MATH  Google Scholar 

  2. Bartoszynski T.: Invariants of measure and category. In: Akihiro, K., Foreman, M. (eds) Handbook of Set-Theory, pp. 491–555. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  3. Blass A., Shelah S.: Ultrafilters with small generating sets. Israel J. Math. 65, 259–271 (1984)

    Article  MathSciNet  Google Scholar 

  4. Brendle J.: Larger cardinals in Cichon’s diagram. J. Symb. Log. 56(3), 795–810 (1991)

    MathSciNet  MATH  Google Scholar 

  5. Brendle J., Fischer V.: Mad families, splitting families and large continuum. J. Symb. Log. 76(1), 198–208 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Goldstern, M.: Tools for your forcing construction. In: Haim, J. (ed.) Set Theory of the Reals. Israel Mathematical Conference Proceedings, pp. 305–360. Bar  Ilan University   (1992)

  7. Jech T.: Set Theory. 3rd millenium edn. Springer, Heidelberg (2002)

    Google Scholar 

  8. Judah H., Shelah S.: The Kunen-Miller chart (Lebesgue measure, the Baire property, Laver reals and preservation theorems for forcing). J. Symb. Log. 55(3), 909–927 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kamburelis A.: Iterations of boolean algebras with measure. Arch. Math. Log. 29, 21–28 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kunen K.: Set Theory: An Introduction to Independence Proofs. North-Holland, Amsterdam (1980)

    MATH  Google Scholar 

  11. Miller A.: Some properties of measure and category. Trans. Am. Math. Soc. 266, 93–114 (1981)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Graduate School of System Informatics, Kobe University, Kobe, Japan

    Diego Alejandro Mejía

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  1. Diego Alejandro Mejía
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Correspondence to Diego Alejandro Mejía.

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Supported by the Monbukagakusho (Ministry of Education, Culture, Sports, Science and Technology) Scholarship, Japan.

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Mejía, D.A. Matrix iterations and Cichon’s diagram. Arch. Math. Logic 52, 261–278 (2013). https://doi.org/10.1007/s00153-012-0315-6

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  • DOI: https://doi.org/10.1007/s00153-012-0315-6

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