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.
Similar content being viewed by others
References
A. Blass, M. Hrusak and J. Verner, On strong P-points, Proceedings of the American Mathematical Society 141 (2013), 2875–2883.
T. Bartošzyński and H. Judah, Set Theory, A K Peters, Wellesley, MA, 1995.
A. Blass, The Rudin-Keisler ordering of P-points, Transactions of the American Mathematical Society 179 (1973), 145–166.
A. Blass, Combinatorial cardinal characteristics of the continuum, in Handbook of Set Theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 395–489.
D. Booth, Ultrafilters on a countable set, Annals of Mathematical Logic 2 (1970/1971), 1–24.
J. Brendle, Strolling through paradise, Fundamenta Mathematicae 148 (1995), 1–25.
J. E. Baumgartner and A. D. Taylor, Partition theorems and ultrafilters, Transactions of the American Mathematical Society 241 (1978), 283–309.
P. E. Cohen, P-points in random universes, Proceedings of the American Mathematical Society 74 (1979), 318–321.
J. Cichoń, A. Rosłanowski, J. Steprāns and B. Weglorz, Combinatorial properties of the ideal β2, Journal fo Symbolic Logic 58 (1993), 42–54.
N. Dobrinen, Survey on the Tukey theory of ultrafilters, Zbornik Radova (Beograd) 17(25) (2015), 53–80.
A. Dow, P-filters and Cohen, Random, and Laver forcing, preprint
C. A. Di Prisco and J. M. Henle, Doughnuts, floating ordinals, square brackets, and ultraflitters, Journal of Symbolic Logic 65 (2000), 461–473.
N. Dobrinen and S. Todorcevic, Tukey types of ultrafilters, Illinois Journal of Mathematics 55 (2011), 907–951.
D. Fernandez-Breton, Generized pathways, unpublished note, https://arxiv.org/abs/1810.06093.
D. Fernandez-Breton and M. Hrusak, Corrigendum to “Gruff ultrafilters” [Topol. Appl. 210 (2016) 355-365], Topology and its Applications 231 (2017), 430–431.
Z. Frolík, Sums of ultrafilters, Bulletin of the American Mathematical Society 73 (1967), 87–91.
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.
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.
L. J. Halbeisen, Combinatorial Set Theory, Springer Monographs in Mathematics, Springer, Cham, 2017.
J. R. Isbell, The category of cofinal types. II, Transactions of the American Mathematical Society 116 (1965), 394–416.
J. Ketonen, On the existence of P-points in the Stone-Cech compactification of integers, Fundamenta Mathematicae 92 (1976), 91–94.
P. Koszmider, A formalism for some class of forcing notions, Zeitschrift fur Math-ematische Logik und Grundlagen der Mathematik 38 (1992), 413–421.
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.
A. R. D. Mathias, Surrealist landscape with figures (a survey of recent results in set theory), Periodica Mathematica Hungarica 10 (1979), 109–175.
J. T. Moore, M. Hrusak and M. Džamonja, Parametrized ◊ principles, Transactions of the American Mathematical Society 356 (2004), 2281–2306.
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.
W. Rudin, Homogeneity problems in the theory of Cech compactifications, Duke Mathematical Journal 23 (1956), 409–419.
S. Shelah, Proper and Improper Forcing, Perspectives in Mathematical Logic, Springer, Berlin, 1998.
J. L. Verner, Lonely points revisited, Commentationes Mathematicae Universitatis Carolinae 54 (2013), 105–110.
J. van Mill, Sixteen topological types in βω - ω, Topology and its Applications 13 (1982), 43–57.
E. L. Wimmers, The Shelah P-point independence theorem, Israel Journal of Mathematics 43 (1982), 28–48.
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.
J. Zapletal, Preserving P-points in definable forcing, Fundamenta Mathematicae 204 (2009), 145–154.
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
Corresponding author
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
About this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-019-1886-2