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There are no P-points in Silver extensions

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.

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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.

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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.

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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

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