Skip to main content
SpringerLink
Log in

There are no P-points in Silver extensions

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We prove that after adding a Silver real no ultrafilter from the ground model can be extended to a P-point, and this remains to be the case in any further extension which has the Sacks property. We conclude that there are no P-points in the Silver model. In particular, it is possible to construct a model without P-points by iterating Borel partial orders. This answers a question of Michael Hrusak. We also show that the same argument can be used for the side-by-side product of Silver forcing. This provides a model without P-points with the continuum arbitrary large, answering a question of Wolfgang Wohofsky.

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. A. Blass, M. Hrusak and J. Verner, On strong P-points, Proceedings of the American Mathematical Society 141 (2013), 2875–2883.

    Google Scholar 

  2. T. Bartošzyński and H. Judah, Set Theory, A K Peters, Wellesley, MA, 1995.

    Book  MATH  Google Scholar 

  3. A. Blass, The Rudin-Keisler ordering of P-points, Transactions of the American Mathematical Society 179 (1973), 145–166.

    Google Scholar 

  4. A. Blass, Combinatorial cardinal characteristics of the continuum, in Handbook of Set Theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 395–489.

    Article  MathSciNet  MATH  Google Scholar 

  5. D. Booth, Ultrafilters on a countable set, Annals of Mathematical Logic 2 (1970/1971), 1–24.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Brendle, Strolling through paradise, Fundamenta Mathematicae 148 (1995), 1–25.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. E. Baumgartner and A. D. Taylor, Partition theorems and ultrafilters, Transactions of the American Mathematical Society 241 (1978), 283–309.

    Google Scholar 

  8. P. E. Cohen, P-points in random universes, Proceedings of the American Mathematical Society 74 (1979), 318–321.

    MathSciNet  MATH  Google Scholar 

  9. J. Cichoń, A. Rosłanowski, J. Steprāns and B. Weglorz, Combinatorial properties of the ideal β2, Journal fo Symbolic Logic 58 (1993), 42–54.

    Article  MathSciNet  MATH  Google Scholar 

  10. N. Dobrinen, Survey on the Tukey theory of ultrafilters, Zbornik Radova (Beograd) 17(25) (2015), 53–80.

    MathSciNet  MATH  Google Scholar 

  11. A. Dow, P-filters and Cohen, Random, and Laver forcing, preprint

  12. C. A. Di Prisco and J. M. Henle, Doughnuts, floating ordinals, square brackets, and ultraflitters, Journal of Symbolic Logic 65 (2000), 461–473.

    Article  MathSciNet  MATH  Google Scholar 

  13. N. Dobrinen and S. Todorcevic, Tukey types of ultrafilters, Illinois Journal of Mathematics 55 (2011), 907–951.

    Article  MathSciNet  MATH  Google Scholar 

  14. D. Fernandez-Breton, Generized pathways, unpublished note, https://arxiv.org/abs/1810.06093.

  15. D. Fernandez-Breton and M. Hrusak, Corrigendum to “Gruff ultrafilters” [Topol. Appl. 210 (2016) 355-365], Topology and its Applications 231 (2017), 430–431.

    Article  MathSciNet  MATH  Google Scholar 

  16. Z. Frolík, Sums of ultrafilters, Bulletin of the American Mathematical Society 73 (1967), 87–91.

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Geschke and S. Quickert, On Sacks forcing and the Sacks property, in Classical and New Paradigms of Computation and their Complexity Hierarchies, Trends in Logic-Studia Logica Library, Vol. 23, Kluwer, Dordrecht, 2004, pp. 95–139.

    MathSciNet  Google Scholar 

  18. O. Guzman, P-points, mad families and cardinal invariants, Ph.D. thesis, Univer-sidad Nacional Autonoma de Mexico, 2017, https://doi.org/abs/1810.09680.

    Google Scholar 

  19. L. J. Halbeisen, Combinatorial Set Theory, Springer Monographs in Mathematics, Springer, Cham, 2017.

    Book  MATH  Google Scholar 

  20. J. R. Isbell, The category of cofinal types. II, Transactions of the American Mathematical Society 116 (1965), 394–416.

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Ketonen, On the existence of P-points in the Stone-Cech compactification of integers, Fundamenta Mathematicae 92 (1976), 91–94.

    Article  MathSciNet  MATH  Google Scholar 

  22. P. Koszmider, A formalism for some class of forcing notions, Zeitschrift fur Math-ematische Logik und Grundlagen der Mathematik 38 (1992), 413–421.

    Article  MathSciNet  MATH  Google Scholar 

  23. K. Kunen, Weak P-points in N*, in Topology, Vol. II (Proc. Fourth Colloq., Budapest, 1978), Colloquia Mathematica Societatis Janos Bolyai, Vol. 23, North-Holland, Amsterdam-New York, 1980, pp. 741–749.

    MATH  Google Scholar 

  24. A. R. D. Mathias, Surrealist landscape with figures (a survey of recent results in set theory), Periodica Mathematica Hungarica 10 (1979), 109–175.

    Article  MathSciNet  MATH  Google Scholar 

  25. J. T. Moore, M. Hrusak and M. Džamonja, Parametrized ◊ principles, Transactions of the American Mathematical Society 356 (2004), 2281–2306.

    Article  MathSciNet  MATH  Google Scholar 

  26. D. Raghavan and S. Shelah, On embedding certain partial orders into the P-points under Rudin-Keisler and Tukey reducibility, Transactions of the American Mathematical Society 369 (2017), 4433–4455.

    Article  MathSciNet  MATH  Google Scholar 

  27. W. Rudin, Homogeneity problems in the theory of Cech compactifications, Duke Mathematical Journal 23 (1956), 409–419.

    Article  MathSciNet  MATH  Google Scholar 

  28. S. Shelah, Proper and Improper Forcing, Perspectives in Mathematical Logic, Springer, Berlin, 1998.

    Google Scholar 

  29. J. L. Verner, Lonely points revisited, Commentationes Mathematicae Universitatis Carolinae 54 (2013), 105–110.

    MathSciNet  MATH  Google Scholar 

  30. J. van Mill, Sixteen topological types in βω - ω, Topology and its Applications 13 (1982), 43–57.

    Article  MathSciNet  Google Scholar 

  31. E. L. Wimmers, The Shelah P-point independence theorem, Israel Journal of Mathematics 43 (1982), 28–48.

    Article  MathSciNet  MATH  Google Scholar 

  32. W. Wohofsky, On the existence of p-points and other ultrafilters in the Stone- Cech-compactification of N, Master's thesis, Vienna University of Technology, 2008.

    Google Scholar 

  33. J. Zapletal, Preserving P-points in definable forcing, Fundamenta Mathematicae 204 (2009), 145–154.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Jindřich Zapletal for multiple inspiring conversations and for suggesting the argument used to prove Theorem 7. The authors would also like to thank Michael Hrušák and Jonathan Verner for valuable discussions on the subject.

Author information

Authors and Affiliations

  1. Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, Praha 1, Czech Republic

    David Chodounský

  2. University of Toronto, 27 King’s College Circle, Toronto, Ontario, M5S 1A1, Canada

    Osvaldo Guzmán

Authors
  1. David Chodounský
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Osvaldo Guzmán
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Chodounský.

Additional information

The first author dedicates this work to his teacher, mentor and dear friend Bohuslav Balcar. The crucial result was proved on the day of his passing.

The first author was supported by the GACR project 17-33849L and RVO: 67985840.

The second author was supported by NSERC grant number 455916 and his visit to Prague was funded by the GACR project 15-34700L and RVO: 67985840. Some of the work and consultations with J. Zapletal were conducted during the ESI workshop Current Trends in Descriptive Set Theory held in December 2016 in Vienna.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chodounský, D., Guzmán, O. There are no P-points in Silver extensions. Isr. J. Math. 232, 759–773 (2019). https://doi.org/10.1007/s11856-019-1886-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-019-1886-2

Search

Navigation