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Projective absoluteness for Sacks forcing

Abstract

We show that \({{\bf \Sigma}^1_3}\)-absoluteness for Sacks forcing is equivalent to the non-existence of a \({{\bf \Delta}^1_2}\) Bernstein set. We also show that Sacks forcing is the weakest forcing notion among all of the preorders that add a new real with respect to \({{\bf \Sigma}^1_3}\) forcing absoluteness.

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Acknowledgments

The author would like to thank Joan Bagaria, through whose lecture in Kobe he became interested in forcing absoluteness. He also appreciates the help given by Yasuo Yoshinobu, who gave him useful comments on this paper and made the proof of Theorem 3.1 very simple. He is grateful to Jörg Brendle for helpful comments on Remark 3.2. Finally, he thanks the referee for several comments on this paper. This research was supported by a GLoRiClass fellowship funded by the European Commission (Early Stage Research Training Mono-Host Fellowship MEST-CT-2005-020841).

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Correspondence to Daisuke Ikegami.

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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Ikegami, D. Projective absoluteness for Sacks forcing. Arch. Math. Logic 48, 679–690 (2009). https://doi.org/10.1007/s00153-009-0143-5

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Keywords

  • Forcing absoluteness
  • Sacks forcing
  • Bernstein sets

Mathematics Subject Classification (2000)

  • 03E15
  • 28A05
  • 54H05