Silver-Like Forcing Notions

Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

On the one hand, we have seen that every forcing notion which adds dominating reals also adds splitting reals (see Fact 20.1). On the other hand, we have seen in the previous chapter that Cohen forcing is a forcing notion which adds splitting reals, but which does not add dominating reals. However, Cohen forcing adds unbounded reals and as an application we constructed a model in which \(\mathfrak {s}=\mathfrak {b}<\mathfrak {d}=\mathfrak {r}\). One might ask whether there exists a forcing notion which is even ω ω-bounding but still adds splitting reals. In this chapter, we shall present such a forcing notion and as an application we shall construct a model in which \(\mathfrak {s}=\mathfrak {b}=\mathfrak {d}<\mathfrak {r}\).

References

  1. 1.
    James E. Baumgartner, Richard Laver: Iterated perfect-set forcing. Ann. Math. Log. 17, 391–397 (1979) MathSciNetCrossRefGoogle Scholar
  2. 2.
    Andreas Blass: Combinatorial cardinal characteristics of the continuum. In: Handbook of Set Theory, vol. 1, Matthew Foreman, Akihiro Kanamori (eds.), pp. 395–490. Springer, Berlin (2010) CrossRefGoogle Scholar
  3. 3.
    Jörg Brendle: Combinatorial properties of classical forcing notions. Ann. Pure Appl. Log. 73, 143–170 (1995) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Jörg Brendle: Strolling through paradise. Fundam. Math. 148, 1–25 (1995) MathSciNetMATHGoogle Scholar
  5. 5.
    Jörg Brendle: How small can the set of generics be? In: Logic Colloquium ’98 (Prague), S.R. Buss, P. Hájek, P. Pudlák (eds.). Lecture Notes in Logic, vol. 13, pp. 109–126. Association for Symbolic Logic, Urbana (2000) Google Scholar
  6. 6.
    Jörg Brendle, Benedikt Löwe: Silver measurability and its relation to other regularity properties. Math. Proc. Camb. Philos. Soc. 138, 135–149 (2005) MATHCrossRefGoogle Scholar
  7. 7.
    Jörg Brendle, Benedikt Löwe: Solovay-type characterizations for forcing-algebras. J. Symb. Log. 64, 1307–1323 (1999) MATHCrossRefGoogle Scholar
  8. 8.
    Jörg Brendle, Shunsuke Yatabe: Forcing indestructibility of MAD families. Ann. Pure Appl. Log. 132, 271–312 (2005) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Stefan Geschke, Sandra Quickert: On Sacks forcing and the Sacks property. In: Classical and New Paradigms of Computation and Their Complexity Hierarchies, Benedikt Löwe, Wolfgang Malzkorn, Thoralf Räsch (eds.). Papers of the Conference Foundations of the Formal Sciences III held in Vienna, September 21–24, 2001. Trends in Logic, vol. 23, pp. 95–139. Kluwer Academic, Dordrecht (2004) CrossRefGoogle Scholar
  10. 10.
    Serge Grigorieff: Combinatorics on ideals and forcing. Ann. Math. Log. 3, 363–394 (1971) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Lorenz Halbeisen: A playful approach to Silver and Mathias forcings. In: Foundations of the Formal Sciences V: Infinite Games, Stefan Bold, Benedikt Löwe, Thoralf Räsch, Johan van Benthem (eds.). Papers of a Conference held in Bonn, November 26–29, 2004. Studies in Logic, vol. 11, pp. 123–142. College Publications, London (2007) Google Scholar
  12. 12.
    Thomas Jech: Multiple Forcing. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1986) MATHGoogle Scholar
  13. 13.
    Jakob Kellner, Saharon Shelah: A Sacks real out of nowhere. J. Symb. Log. 75, 51–75 (2010) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Miloš S. Kurilić: Splitting families and forcing. Ann. Pure Appl. Log. 145, 240–251 (2007) MATHCrossRefGoogle Scholar
  15. 15.
    Richard Laver: Products of infinitely many perfect trees. J. Lond. Math. Soc. (2) 29, 385–396 (1984) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Adrian Richard David Mathias: Surrealist landscape with figures (a survey of recent results in set theory). Period. Math. Hung. 10, 109–175 (1979) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Arnold W. Miller: Rational perfect set forcing. In: Axiomatic Set Theory, James E. Baumgartner, Donald A. Martin, Saharon Shelah (eds.). Contemporary Mathematics, vol. 31, pp. 143–159. Am. Math. Soc., Providence (1984) CrossRefGoogle Scholar
  18. 18.
    Andrzej Rosłanowski, Juris Steprāns: Chasing Silver. Can. Math. Bull. 51, 593–603 (2008) MATHCrossRefGoogle Scholar
  19. 19.
    Gerald E. Sacks: Forcing with Perfect Closed Sets. In: Axiomatic Set Theory, Dana S. Scott (eds.). Proceedings of Symposia in Pure Mathematics, vol. XIII, Part I, pp. 331–355. Am. Math. Soc., Providence (1971) CrossRefGoogle Scholar
  20. 20.
    Petr Simon: Sacks forcing collapses \(\mathfrak {c}\) to \(\mathfrak {b}\). Comment. Math. Univ. Carol. 34, 707–710 (1993) MATHGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Institut für MathematikUniversität ZürichZürichSwitzerland

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