Construction with opposition: cardinal invariants and games

  • Jörg Brendle
  • Michael Hrušák
  • Víctor Torres-PérezEmail author
Open Access


We consider several game versions of the cardinal invariants \({\mathfrak {t}}\), \({\mathfrak {u}}\) and \({\mathfrak {a}}\). We show that the standard proof that parametrized diamond principles prove that the cardinal invariants are small actually shows that their game counterparts are small. On the other hand we show that \({\mathfrak {t}}<{\mathfrak {t}}_{Builder}\) and \({\mathfrak {u}}<{\mathfrak {u}}_{Builder}\) are both relatively consistent with ZFC, where \({\mathfrak {t}}_{Builder}\) and \({\mathfrak {u}}_{Builder}\) are the principal game versions of \({\mathfrak {t}}\) and \({\mathfrak {u}}\), respectively. The corresponding question for \({\mathfrak {a}}\) remains open.


Cardinal invariants of the continuum Transfinite games Parametrized diamond principles 

Mathematics Subject Classification

03E05 03E15 03E17 



Open access funding provided by Austrian Science Fund (FWF).


  1. 1.
    Balcar, B., Doucha, M., Hrušák, M.: Base tree property. Order 32(1), 69–81 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Balcar, B., Pelant, J., Simon, P.: The space of ultrafilters on \({\mathbb{N}}\) covered by nowhere dense sets. Fund. Math. 110, 11–24 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bartoszyński, T., Judah, H.: Set Theory. On the Structure of the Real Line. A K Peters, Wellesley (1995)CrossRefzbMATHGoogle Scholar
  4. 4.
    Baumgartner, J.E., Dordal, P.: Adjoining dominating functions. J. Symb. Log. 50(1), 94–101 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Blass, A.: Combinatorial cardinal characteristics of the continuum. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set theory, pp. 395–489. Springer, Berlin (2009)Google Scholar
  6. 6.
    Brendle, J., Raghavan, D.: Bounding, splitting, and almost disjointness. Ann. Pure Appl. Log. 165(2), 631–651 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brendle, J., Shelah, S.: Ultrafilters on \(\omega \)—their ideals and their cardinal characteristics. Trans. Am. Math. Soc. 351(7), 2643–2674 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Devlin, K.J., Shelah, S.: A weak version of \(\diamondsuit \) which follows from \(2^{\aleph _0}<2^{\aleph _1}\). Israel J. Math. 29(2), 239–247 (1978)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dow, A.: More set-theory for topologists. Topol. Appl. 64(3), 243–300 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Foreman, M.: Games played on Boolean algebras. J. Symb. Log. 48(3), 714–723 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jensen, R.B.: The fine structure of the constructible hierarchy. Ann. Math. Log. 4, 229–308 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Judah, H., Shelah, S.: \(\varvec {\Delta }^1_2\)-sets of reals. Ann. Pure Appl. Log. 42, 207–223 (1989)CrossRefGoogle Scholar
  13. 13.
    Judah, H., Shelah, S.: Q-sets, Sierpinski sets, and rapid filters. Proc. Am. Math. Soc. 111, 821–832 (1991)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Malliaris, M., Shelah, S.: Cofinality spectrum theorems in model theory, set theory, and general topology. J. Am. Math. Soc. 29(1), 237–297 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Moore, J.T., Hrušák, M., Džamonja, M.: Parametrized \(\diamondsuit \)-principles. Trans. Am. Math. Soc. 356(6), 2281–2306 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Shelah, S.: Proper and Improper Forcing, Perspectives in Mathematical Logic, 2nd edn. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  17. 17.
    Vojtáš, P.: Game properties of Boolean algebras. Comment. Math. Univ. Carol. 24(2), 349–369 (1983)MathSciNetzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, 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.

Authors and Affiliations

  1. 1.Graduate School of System InformaticsKobe UniversityKobeJapan
  2. 2.Centro de Ciencias MatemáticasUNAMMoreliaMexico
  3. 3.Institute of Discrete Mathematics and GeometryTU WienViennaAustria

Personalised recommendations