Skip to main content
Log in

Ideals on \({P_{\kappa}(\lambda)}\) associated with games of uncountable length

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

Abstract

We study normal ideals on \({P_{\kappa} (\lambda)}\) that are defined in terms of games of uncountable length.

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. Abraham U., Magidor M. : Cardinal arithmetic. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set Theory, vol. 2, pp. 1149–1227. Springer, Berlin (2010)

    Chapter  Google Scholar 

  2. Cummings J., Foreman M., Magidor M.: Squares, scales and stationary reflection. J. Math. Log. 1, 35–98 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cummings J., Foreman M., Magidor M.: Canonical structure in the universe of set theory I. Ann. Pure Appl. Log. 129, 211–243 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Dobrinen N.: κ-stationary subsets of \({\mathcal P_{\kappa^+} \lambda}\) , infinitary games, and distributive laws in Boolean algebras. J. Symb. Log. 73, 238–260 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Donder, H.D., Levinski, J.P.: Stationary sets and game filters, ca. (1987) (unpublished)

  6. Donder H.D., Matet P.: Two cardinal versions of diamond. Isr. J. Math. 83, 1–43 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  7. Eisworth T.: Successors of singular cardinals. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set Theory, vol. 2, pp. 1229–1350. Springer, Berlin (2010)

    Chapter  Google Scholar 

  8. Foreman M.: Potent axioms. Trans. Am. Math. Soc. 294, 1–28 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  9. Foreman M., Magidor M.: A very weak square principle. J. Symb. Log. 62, 175–196 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  10. Foreman M., Magidor M.: Mutually stationary sequences of sets and the non-saturation of the non-stationary ideal on \({P_{\kappa}(\lambda)}\) . Acta Math. 186, 271–300 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. Foreman M., Todorcevic S.: A new Löwenheim–Skolem theorem. Trans. Am. Math. Soc. 357, 1693–1715 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gitik M., Sharon A.: On SCH and the approachability property. Proc. Am. Math. Soc. 136, 311–320 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gitik M., Shelah S.: Less saturated ideals. Proc. Am. Math. Soc. 125, 1523–1530 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  14. Holz M., Steffens K., Weitz E.: Introductrion to Cardinal Arithmetic, Birkhäuser Advanced Texts : Basler Lehrbücher. Birkhäuser, Basel (1999)

    Google Scholar 

  15. Huuskonen T., Hyttinen T., Rautila M.: On the κ-cub game on λ and I[λ]. Arch. Math. Log. 38, 549–557 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  16. Huuskonen T., Hyttinen T., Rautila M.: On potential isomorphism and non-structure. Arch. Math. Log. 43, 85–120 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Jech T.J.: Set Theory, The Third Millenium Edition, Springer monographs in Mathematics. Springer, Berlin (2002)

    Google Scholar 

  18. Kueker D.W.: Countable approximations and Löwenheim–Skolem theorems. Ann. Math. Log. 11, 57–103 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  19. Matet, P.: Concerning stationary subsets of \({[\lambda]^{<\kappa}}\) . In: Steprāns, J., Watson, S. (eds.) Set Theory and Its Applications, Lecture Notes in Mathematics, vol. 1401, pp. 119–127. Springer, Berlin (1989)

  20. Matet P.: Large cardinals and covering numbers. Fundam. Math. 205, 45–75 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  21. Matet P.: Game ideals. Ann. Pure Appl. Log. 158, 23–39 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  22. Matet P.: Weak saturation of ideals on \({P_{\kappa} (\lambda)}\) . Math. Log. Q. 57, 149–165 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  23. Matet P.: Non-saturation of the non-stationary ideal on \({P_{\kappa} (\lambda)}\) with λ of countable cofinality. Math. Log. Q. 58, 38–45 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  24. Matet P.: Non-saturation of the non-stationary ideal on \({P_{\kappa} (\lambda)}\) in case \({\kappa\leq {\rm cf} (\lambda) < \lambda}\) . Arch. Math. Log. 51, 425–432 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  25. Matet P.: Normal restrictions of the noncofinal ideal on \({P_{\kappa} (\lambda)}\) . Fundam. Math. 221, 1–22 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  26. Matet, P.: Two-cardinal diamond and games of uncountable length (preprint)

  27. Matet P., Péan C., Shelah S.: Cofinality of normal ideals on \({P_{\kappa}(\lambda)}\) II. Isr. J. Math. 150, 253–283 (2005)

    Article  MATH  Google Scholar 

  28. Matet, P., Péan, C., Shelah, S.: Cofinality of normal ideals on \({P_{\kappa}(\lambda)}\) I (preprint)

  29. Matet, P., Shelah, S.: The nonstationary ideal on \({P_{\kappa}(\lambda)}\) for λ singular (preprint)

  30. Mekler A.H., Shelah S.: Stationary logic and its friends. II. Notre Dame J. Form. Log. 27, 39–50 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  31. Mitchell W.J.: I2] can be the nonstationary ideal on \({{\rm Cof} (\omega_1)}\) . Trans. Am. Math. Soc. 361, 561–601 (2009)

    Article  MATH  Google Scholar 

  32. Shelah S.: Cardinal Arithmetic, Oxford Logic Guides, vol. 29. Oxford University Press, Oxford (1994)

    Google Scholar 

  33. Solovay, R.M.: Real-valued measurable cardinals. In: Scott, D.S. (ed.) Axiomatic Set Theory, Proceedings of Symposia in Pure Mathematcis, vol. 13, part 1, American Mathematical Society, Providence, pp. 397–428 (1971)

  34. Usuba T.: Splitting stationary sets in \({\mathcal{P}_\kappa\lambda}\) for λ with small cofinality. Fundam. Math. 205, 265–287 (2009)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Pierre Matet.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Matet, P. Ideals on \({P_{\kappa}(\lambda)}\) associated with games of uncountable length. Arch. Math. Logic 54, 291–328 (2015). https://doi.org/10.1007/s00153-014-0412-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-014-0412-9

Keywords

Mathematics Subject Classification

Navigation