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Creature forcing and five cardinal characteristics in Cichoń’s diagram

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  • Published: 03 June 2017
  • Volume 56, pages 1045–1103, (2017)
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Creature forcing and five cardinal characteristics in Cichoń’s diagram
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  • Arthur Fischer1,
  • Martin Goldstern2,
  • Jakob Kellner2 &
  • …
  • Saharon Shelah3,4 
  • 739 Accesses

  • 11 Citations

  • 10 Altmetric

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Abstract

We use a (countable support) creature construction to show that consistently

$$\begin{aligned} \mathfrak d=\aleph _1= {{\mathrm{cov}}}(\mathcal N)< {{\mathrm{non}}}(\mathcal M)< {{\mathrm{non}}}(\mathcal N)< {{\mathrm{cof}}}(\mathcal N) < 2^{\aleph _0}. \end{aligned}$$

The same method shows the consistency of

$$\begin{aligned} \mathfrak d=\aleph _1= {{\mathrm{cov}}}(\mathcal N)< {{\mathrm{non}}}(\mathcal N)< {{\mathrm{non}}}(\mathcal M)< {{\mathrm{cof}}}(\mathcal N) < 2^{\aleph _0}. \end{aligned}$$

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References

  1. Bartoszyński, T.: Additivity of measure implies additivity of category. Trans. Am. Math. Soc. 281(1), 209–213 (1984)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Bartoszyński, T., Judah, H.: Set Theory. On the Structure of the Real Line. A K Peters Ltd., Wellesley (1995)

    MATH  Google Scholar 

  4. Bartoszyński, T., Judah, H., Shelah, S.: The Cichoń diagram. J. Symb. Log. 58(2), 401–423 (1993)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Blass, A., Shelah, S.: Ultrafilters with small generating sets. Israel J. Math. 65(3), 259–271 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cichoń, J., Kamburelis, A., Pawlikowski, J.: On dense subsets of the measure algebra. Proc. Am. Math. Soc. 94(1), 142–146 (1985)

    Article  MathSciNet  MATH  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(1), 21–28 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. Krawczyk, A.: Consistency of A(c) & B(m) & non-A(m). Unpublished notes (1983)

  11. Kellner, J., Shelah, S.: Decisive creatures and large continuum. J. Symb. Log. 74(1), 73–104 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kellner, J., Shelah, S.: Creature forcing and large continuum: the joy of halving. Arch. Math. Log. 51(1–2), 49–70 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mejía, D.A.: Matrix iterations and Cichon’s diagram. Arch. Math. Log. 52(3–4), 261–278 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Miller, A.W.: Some properties of measure and category. Trans. Am. Math. Soc. 266(1), 93–114 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  15. Miller, A.W.: Additivity of measure implies dominating reals. Proc. Am. Math. Soc. 91(1), 111–117 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  16. Raisonnier, J., Stern, J.: Mesurabilité et propriété de Baire. C. R. Acad. Sci. Paris Sér. I Math. 296(7), 323–326 (1983)

    MathSciNet  MATH  Google Scholar 

  17. Rosłanowski, A., Shelah, S.: Norms on possibilities. I. Forcing with trees and creatures. Mem. Am. Math. Soc. 141(671), xii+167 (1999)

    MathSciNet  MATH  Google Scholar 

  18. Shelah, S.: Proper and Improper Forcing. Perspectives in Mathematical Logic, 2nd edn. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

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Acknowledgements

Open access funding provided by Austrian Science Fund (FWF). We are grateful to Diego Mejía for pointing out several embarrassing oversights. We also thank the anonymous referee for pointing out additional errors, and making numerous helpful suggestions for improving the text.

Author information

Authors and Affiliations

  1. Kurt Gödel Research Center for Mathematical Logic, Universität Wien, Währinger Straße 25, 1090, Vienna, Austria

    Arthur Fischer

  2. Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8–10/104, 1040, Vienna, Austria

    Martin Goldstern & Jakob Kellner

  3. Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, 91904, Jerusalem, Israel

    Saharon Shelah

  4. Department of Mathematics, Rutgers University, New Brunswick, NJ, 08854, USA

    Saharon Shelah

Authors
  1. Arthur Fischer
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  2. Martin Goldstern
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Corresponding author

Correspondence to Jakob Kellner.

Additional information

Dedicated to the memory of James E. Baumgartner (1943–2011).

We gratefully acknowledge the following partial support: Austrian Science Fund FWF P23875-N13 (first author), P24725-N25 (second author) and I1272-N25 (third author); US National Science Foundation NSF DMS-1362974 (second author), and European Research Council grant ERC-2013-ADG 338821 (fourth author). This is publication 1044 of the fourth author.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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Fischer, A., Goldstern, M., Kellner, J. et al. Creature forcing and five cardinal characteristics in Cichoń’s diagram. Arch. Math. Logic 56, 1045–1103 (2017). https://doi.org/10.1007/s00153-017-0553-8

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  • Received: 15 January 2014

  • Accepted: 22 May 2015

  • Published: 03 June 2017

  • Issue date: November 2017

  • DOI: https://doi.org/10.1007/s00153-017-0553-8

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Keywords

  • Set theory of the reals
  • Creature forcing
  • Cichoń’s diagram

Mathematics Subject Classification

  • 03E17
  • 03E35
  • 03E40

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