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Saturation, Suslin trees and meager sets

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Abstract.

We show, using a variation of Woodin’s partial order ℙ max , that it is possible to destroy the saturation of the nonstationary ideal on ω1 by forcing with a Suslin tree. On the other hand, Suslin trees typcially preserve saturation in extensions by ℙ max variations where one does not try to arrange it otherwise. In the last section, we show that it is possible to have a nonmeager set of reals of size ℵ1, saturation of the nonstationary ideal, and no weakly Lusin sequences, answering a question of Shelah and Zapletal.

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References

  1. Baumgartner, J., Taylor, A.: Saturation Properties of Ideals in Generic Extensions, II. Trans. Amer. Math. Soc. 271 (2), 587–609 (1982)

    Google Scholar 

  2. Bartoszyński, T., Judah, H.: Set theory. On the structure of the real line, A K Peters, Ltd., Wellesley, MA, 1995

  3. Jech, T., Magidor, M., Mitchell, W., Prikry, K.: Precipitous ideals. J. Symbolic Logic 45 (1), 1–8 (1980)

    Google Scholar 

  4. Kakuda, Y.: On a condition for Cohen extensions which preserve precipitous ideals. J. Symbolic Logic 46 (2), 296–300 (1981)

    Google Scholar 

  5. Kanamori, A.: The higher infinite. Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1994

  6. Larson, P.: An variation for one Souslin tree. J. Symbolic Logic 64, 81–98 (1999)

    Google Scholar 

  7. Larson, P.: Forcing over models of determinacy. To appear in the Handbook of Set Theory, Foreman, Kanamori, Magidor, eds

  8. Larson, P., Todorčević, S.: Chain conditions in maximal models. Fund. Math. 168 (1), 77–104 (2001)

    Google Scholar 

  9. Laver, R.: Saturated ideals and nonregular ultrafilters. In: G. Metakides (ed.), Patras Logic Symposion (Patras 1980), North-Holland, 1982, pp. 297–305

  10. Magidor, M.: Precipitous ideals and Σ41 sets. Israel J. Math. 35 (1–2), 109–134 (1980)

    Google Scholar 

  11. Seabold, D.: Chang’s conjecture and the nonstationary ideal. J. Symbolic Logic 66 (1), 144–170 (2001)

    Google Scholar 

  12. Shelah, S., Zapletal, J.: Canonical models for ℵ1-combinatorics. Ann. Pure Appl. Logic 98, 217–259 (1999)

    Article  Google Scholar 

  13. Veličković, B.: Forcing axioms and stationary sets. Adv. Math. 94 (2), 256–284 (1992)

    Article  Google Scholar 

  14. Woodin, W.H.: The axiom of determinacy, forcing axioms, and the nonstationary ideal. DeGruyter Series in Logic and Its Applications, vol. 1, 1999

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Miami University, Oxford, OH, 45056, USA

    Paul Larson

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  1. Paul Larson
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Correspondence to Paul Larson.

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Supported by the Japan Society for the Promotion of Science, the Mittag-Leffler Institute and the São Paulo State Research Support Foundation (FAPESP, Grant # 02/11551-3).

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Larson, P. Saturation, Suslin trees and meager sets. Arch. Math. Logic 44, 581–595 (2005). https://doi.org/10.1007/s00153-004-0257-8

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  • DOI: https://doi.org/10.1007/s00153-004-0257-8

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