Skip to main content
SpringerLink
Log in

Strongly dominating sets of reals

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

Abstract

We analyze the structure of strongly dominating sets of reals introduced in Goldstern et al. (Proc Am Math Soc 123(5):1573–1581, 1995). We prove that for every \({\kappa < \mathfrak{b}}\) a \({\kappa}\) -Suslin set \({A\subseteq{}^\omega\omega}\) is strongly dominating if and only if A has a Laver perfect subset. We also investigate the structure of the class l of Baire sets for the Laver category base and compare the σ-ideal of sets which are not strongly dominating with the Laver ideal l 0.

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

Access this article

Log in via an institution

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., Welleslay, Massachusetts (1995)

    MATH  Google Scholar 

  2. Brendle J., Hjorth G., Spinas O.: Regularity properties for dominating projective sets. Ann. Pure Appl. Log. 72, 291–307 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Goldstern M., Repický M., Shelah S., Spinas O.: On tree ideals. Proc. Am. Math. Soc. 123(5), 1573–1581 (1995)

    Article  MATH  Google Scholar 

  4. Judah H., Miller A., Shelah S.: Sacks forcing, Laver forcing and Martin’s axiom. Arch. Math. Log. 31, 145–161 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Kechris A.S.: On a notion of smallness for subsets of the Baire space. Trans. Am. Math. Soc. 229, 191–207 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kułaga W.: On fields and ideals connected with notions of forcing. Coll. Math. 105(2), 271–281 (2006)

    Article  MATH  Google Scholar 

  7. Miller, A.W.: Hechler and Laver Trees, preprint. http://arxiv.org/abs/1204.5198

  8. Morgan J.C. II: Point Set Theory. Dekker, New York (1990)

    MATH  Google Scholar 

  9. Moschovakis, Y.N.: Descriptive Set Theory, Mathematical Surveys and Monographs, 2nd edn., vol. 155. AMS, Providence, RI (2009)

  10. Repický, M.: Bases of measurability in Boolean algebras. Math. Slovaca, to appear

  11. Spinas O.: Dominating projective sets in the Baire space. Ann. Pure Appl. Log. 68, 327–342 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  12. van Douwen E.K. : The integers and topology, chapter 3. In: Kunen, K., Vaughan, J.E. (eds) Handbook of Set-Theoretic Topology, pp. 111–167. North-Holland, Amsterdam (1984)

    Google Scholar 

  13. Zapletal J.: Isolating cardinal invariants. J. Math. Log. 3(1), 143–162 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, P. J. Šafárik University, Jesenná 5, 041 54, Košice, Slovak Republic

    Michal Dečo

  2. Institute of Mathematics, Slovak Academy of Sciences, Grešákova 6, 040 01, Košice, Slovak Republic

    Miroslav Repický

Authors
  1. Michal Dečo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Miroslav Repický
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Miroslav Repický.

Additional information

The first author was supported by grant VEGA 1/0002/12 and the second author was supported by grants VEGA 1/0002/12 and APVV-0269-11.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dečo, M., Repický, M. Strongly dominating sets of reals. Arch. Math. Logic 52, 827–846 (2013). https://doi.org/10.1007/s00153-013-0347-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-013-0347-6

Keywords

Mathematics Subject Classification (2000)

Search

Navigation