Skip to main content
Log in

Many different covering numbers of Yorioka’s ideals

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract

For \({b \in {^{\omega}}{\omega}}\) , let \({\mathfrak{c}^{\exists}_{b, 1}}\) be the minimal number of functions (or slaloms with width 1) to catch every functions below b in infinitely many positions. In this paper, by using the technique of forcing, we construct a generic model in which there are many coefficients \({\mathfrak{c}^{\exists}_{{b_\alpha}, 1}}\) with pairwise different values. In particular, under the assumption that a weakly inaccessible cardinal exists, we can construct a generic model in which there are continuum many coefficients \({\mathfrak{c}^{\exists}_{{b_\alpha}, 1}}\) with pairwise different values. In conjunction with these results, we give a generic model in which there are many Yorioka’s ideals \({\mathcal{I}_{f_\alpha}}\) with pairwise different covering numbers.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

  2. Goldstern M., Shelah S.: Many simple cardinal coefficients. Arch. Math. Log. 32, 203–221 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  3. Kellner J.: Even more simple cardinal coefficients. Arch. Math. Log. 47, 503–515 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Kunen, K.: Set Theory. North Holland, New York (1980)

  7. Osuga N.: The covering number and the uniformity of the ideal \({{\mathcal{I}_{f_\alpha}}}\) . Math. Log. Q. 52(4), 351–358 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Osuga N., Kamo S.: The cardinal coefficients of the ideal \({{\mathcal{I}_{f_\alpha}}}\) . Arch. Math. Log. 47(7–8), 653–671 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Yorioka T.: The cofinality of the strong measure zero ideal. J. Symb. Log. 67(4), 1373–1384 (2002)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Noboru Osuga.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Osuga, N., Kamo, S. Many different covering numbers of Yorioka’s ideals. Arch. Math. Logic 53, 43–56 (2014). https://doi.org/10.1007/s00153-013-0354-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-013-0354-7

Keywords

Mathematics Subject Classification (2000)

Navigation