Skip to main content
SpringerLink
Log in

Towers and clubs

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

Abstract

We revisit several results concerning club principles and nonsaturation of the nonstationary ideal, attempting to improve them in various ways. So we typically deal with a (non necessarily normal) ideal J extending the nonstationary ideal on a regular uncountable (non necessarily successor) cardinal \(\kappa \), our goal being to witness the nonsaturation of J by the existence of towers (of length possibly greater than \(\kappa ^+\)).

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. Balcar, B., Simon, P.: Disjoint refinement, in : Handbook of Boolean Algebras (J.D. Monk and R. Bonnet, eds.), vol. 2, North-Holland, 1989, pp. 333–388

  2. Baumgartner, J.E.: Almost disjoint sets, the dense set problem and the partition calculus. Ann. Math. Log. 10, 401–439 (1976)

    Article  MathSciNet  Google Scholar 

  3. Baumgartner, J.E., Taylor, A.D., Wagon, S.: Structural properties of ideals, Dissertationes Mathematicae CXCVII, Warszawa, 1982

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

    Article  MathSciNet  Google Scholar 

  5. Cummings, J., Shelah, S.: Cardinal invariants above the continuum. Ann. Pure Appl. Log. 75, 251–268 (1995)

    Article  MathSciNet  Google Scholar 

  6. Džamonja, M.: On club-like principles on regular cardinals above bethw. Acta Universitatis Carolinae - Mathematica et Physica 42, 27–32 (2001)

    MathSciNet  MATH  Google Scholar 

  7. Džamonja, M., Shelah, S.: Saturated filters at successors of singulars, weak reflection and yet another weak club principle. Ann. Pure Appl. Log. 79, 289–316 (1996)

    Article  MathSciNet  Google Scholar 

  8. Džamonja, M., Shelah, S.: Similar but not the same : various versions of \(\clubsuit \) do not coincide. J. Symb. Log. 64, 180–198 (1999)

    Article  MathSciNet  Google Scholar 

  9. Džamonja, M., Shelah, S.: On versions of \(\clubsuit \) on cardinals larger than \(\aleph _1\). Mathematica Japonica 51, 53–61 (2000)

    MathSciNet  MATH  Google Scholar 

  10. Foreman, M.: More saturated ideals, in : Cabal seminar 79-81, Lecture Notes in Mathematics 1019, Springer, 1983, pp. 1–27

  11. 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  MathSciNet  Google Scholar 

  12. Fuchino, S., Shelah, S., Soukup, L.: Sticks and clubs. Ann. Pure Appl. Log. 90, 57–77 (1997)

    Article  MathSciNet  Google Scholar 

  13. Fuchs, G., Rinot, A.: Weak square and stationary reflection. Acta Mathematica Hungarica 155, 393–405 (2018)

    Article  MathSciNet  Google Scholar 

  14. Garti, S.: Tiltan. Comptes Rendus de l’Académie des Sciences de Paris, Série I(356), 351–359 (2018)

    MathSciNet  MATH  Google Scholar 

  15. Gitik, M.: personal communication

  16. Gitik, M., Rinot, A.: The failure of diamond on a reflecting stationary set. Trans. Am. Math. Soc. 364, 1771–1795 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Gregory, J.: Higher Souslin trees and the generalized continuum hypothesis. J. Symb. Log. 41, 663–671 (1976)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  20. Jensen, R.B.: The fine structure of the constructible hierarchy. Ann. Math. Log. 4, 229–308 (1972)

    Article  MathSciNet  Google Scholar 

  21. Kojman, M.: Splitting families of sets in ZFC. Adv. Math. 269, 707–725 (2015)

    Article  MathSciNet  Google Scholar 

  22. König, B., Larson, P., Yoshinobu, Y.: Guessing clubs in the generalized club filter. Fundamenta Mathematicae 195, 177–191 (2007)

    Article  MathSciNet  Google Scholar 

  23. Krueger, J.: Fat sets and saturated ideals. J. Symb. Log. 68, 837–845 (2003)

    Article  MathSciNet  Google Scholar 

  24. Kumar, A., Shelah, S.: On some variants of the club principle, preprint

  25. Kunen, K.: Set Theory, Studies in Logic and the Foundations of Mathematics, vol. 102. North-Holland, Amsterdam (1980)

    Google Scholar 

  26. Matet, P.: Weak saturation of ideals on \(P_\kappa (\lambda )\). Math. Log. Quart. 57, 149–165 (2011)

    Article  MathSciNet  Google Scholar 

  27. Matet, P.: Two-cardinal diamond star. Math. Log. Quart. 60, 246–265 (2014)

    Article  MathSciNet  Google Scholar 

  28. Matet, P.: Normal restrictions of the non-cofinal ideal on \(P_\kappa (\lambda )\). Fundamenta Mathematicae 221, 1–22 (2013)

    Article  MathSciNet  Google Scholar 

  29. Matet, P.: Guessing more sets. Ann. Pure Appl. Log. 166, 953–990 (2015)

    Article  MathSciNet  Google Scholar 

  30. Matet, P. : Meeting numbers and pseudopowers. Mathematical Logic Quarterly (to appear)

  31. Matet, P.: Versions for \(P_\kappa (\lambda )\) of two results of Shelah on diamond - Part 1, preprint

  32. Matsubara, Y., Sakai, H., Usuba, T.: On the existence of skinny stationary subsets. Ann. Pure Appl. Log. 170, 539–557 (2019)

    Article  MathSciNet  Google Scholar 

  33. Merimovich, C.: A power function with a fixed finite gap everywhere. J Symb. Log. 72, 361–417 (2007)

    Article  MathSciNet  Google Scholar 

  34. Mildenberger, H.: Finding generic filters by playing games. Arch. Math. Logic 49, 91–118 (2010)

    Article  MathSciNet  Google Scholar 

  35. Ostaszewski, A.J.: On countably compact perfectly normal spaces. J. Lond. Math. Soc. 2, 505–516 (1975)

    MathSciNet  MATH  Google Scholar 

  36. Primavesi, A.: Guessing Axioms, Invariance and Suslin Trees, Doctoral dissertation, University of East Anglia, 2011

  37. Rinot, A.: A relative of the approachability ideal, diamond and non-saturation. J. Symb. Log. 75, 1035–1065 (2010)

    Article  MathSciNet  Google Scholar 

  38. Rinot, A.: Jensen’s diamond principle and its relatives, in : Set Theory and its Applications (L. Babinkostova et al., eds.), Contemporary Mathematics vol. 533, American Mathematical Society, Providence, RI, 2011, pp. 125-156

  39. Rinot, A.: On guessing generalized clubs at the successors of regulars. Ann. Pure Appl. Log. 162, 566–577 (2011)

    Article  MathSciNet  Google Scholar 

  40. Shelah, S.: On successors of singular cardinals, in : Logic Colloquium ’78 (M. Boffa et al., eds.), Studies in Logic and the Foundations of Mathematics vol. 97, North-Holland, Amsterdam, 1979, pp. 357-380

  41. Shelah, S.: Whitehead groups may not be free, even assuming CH, II. Israel J. Math. 35, 257–285 (1980)

    Article  MathSciNet  Google Scholar 

  42. Shelah, S.: Models with second-order properties. III. Omitting types for \(L(Q)^\ast \), Archiv für Mathematische Logik und Grundlagenforschung 21 (1981), 1–11

  43. Shelah, S.: Advances in cardinal arithmetic, in : Finite and Infinite Combinatorics in Sets and Logic (N. Sauer et al., eds.), Kluwer, Dordrecht, 1993, pp. 355–383

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

    Google Scholar 

  45. Shelah, S.: The generalized continuum hypothesis revisited. Israel J. Math. 116, 285–321 (2000)

    Article  MathSciNet  Google Scholar 

  46. Shelah, S.: Applications of PCF theory. J. Symb. Log. 65, 1624–1674 (2000)

    Article  MathSciNet  Google Scholar 

  47. Shelah, S.: Diamonds. Proc. Am. Math. Soc. 138, 2151–2161 (2010)

    Article  MathSciNet  Google Scholar 

  48. Shioya, M.: Splitting \(\cal{P}_\kappa \lambda \) into maximally many stationary sets. Israel J. Math. 114, 347–357 (1999)

    Article  MathSciNet  Google Scholar 

  49. Tall, F.D.: Some applications of a generalized Martin’s axiom. Topol. Appl. 57, 215–248 (1994)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author would like to thank Assaf Rinot for fruitful discussions. He is deeply indebted to Moti Gitik for his contribution to this paper. Finally, he would like to extend thanks to the referee for a long list of detailed suggestions for improvement.

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques, Université de Caen - CNRS, BP 5186, 14032, Caen Cedex, France

    Pierre Matet

Authors
  1. Pierre Matet
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pierre Matet.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Matet, P. Towers and clubs. Arch. Math. Logic 60, 683–719 (2021). https://doi.org/10.1007/s00153-020-00754-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-020-00754-3

Keywords

Mathematics Subject Classification

Search

Navigation