Advertisement

Cohen Forcing Revisited

  • Lorenz J. Halbeisen
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

Since Cohen forcing is countable, it satisfies ccc, hence, Cohen forcing is proper. Furthermore, since forcing notions with the Laver property do not add Cohen reals, Cohen forcing obviously does not have the Laver property.

Not so obvious are the facts that Cohen forcing adds unbounded and splitting, but no dominating reals.

References

  1. 1.
    Tomek Bartoszyński: Additivity of measure implies additivity of category. Trans. Am. Math. Soc. 281, 209–213 (1984) MATHGoogle Scholar
  2. 2.
    Tomek Bartoszyński: Combinatorial aspects of measure and category. Fundam. Math. 127, 225–239 (1987) MATHGoogle Scholar
  3. 3.
    Tomek Bartoszyński, Haim Judah: Set Theory: On the Structure of the Real Line. AK Peters, Wellesley (1995) MATHGoogle Scholar
  4. 4.
    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
  5. 5.
    Thomas Jech: Multiple Forcing. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1986) MATHGoogle Scholar
  6. 6.
    Thomas Jech: Set Theory, The Third Millennium Edition, Revised and Expanded. Springer Monographs in Mathematics. Springer, Berlin (2003) MATHGoogle Scholar
  7. 7.
    Kenneth Kunen: Some points in βN. Math. Proc. Camb. Philos. Soc. 80, 385–398 (1976) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Kenneth Kunen: Set Theory, an Introduction to Independence Proofs. Studies in Logic and the Foundations of Mathematics, vol. 102. North-Holland, Amsterdam (1983) MATHGoogle Scholar
  9. 9.
    Miloš S. Kurilić: Cohen-stable families of subsets of integers. J. Symb. Log. 66, 257–270 (2001) MATHCrossRefGoogle Scholar
  10. 10.
    Arnold W. Miller: Some properties of measure and category. Trans. Am. Math. Soc. 266, 93–114 (1981) MATHCrossRefGoogle Scholar
  11. 11.
    Janusz Pawlikowski: Why Solovay real produces Cohen real. J. Symb. Log. 51, 957–968 (1986) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Zbigniew Piotrowski, Andrzej Szymański: Some remarks on category in topological spaces. Proc. Am. Math. Soc. 101, 156–160 (1987) MATHCrossRefGoogle Scholar
  13. 13.
    Robert M. Solovay: A model of set theory in which every set of reals is Lebesgue measurable. Ann. Math. (2) 92, 1–56 (1970) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Robert M. Solovay: Real-valued measurable cardinals. In: Axiomatic Set Theory, Dana S. Scott (ed.). Proceedings of Symposia in Pure Mathematics, vol. XIII, Part I, pp. 397–428. Am. Math. Soc., Providence (1971) CrossRefGoogle Scholar
  15. 15.
    Jacques Stern: Partitions of the real line into1 closed sets. In: Higher Set Theory, Proceedings, Oberwolfach, Germany, April 13–23, 1977, Gert H. Müller, Dana S. Scott (eds.). Lecture Notes in Mathematics, vol. 669, pp. 455–460. Springer, Berlin (1978) CrossRefGoogle Scholar
  16. 16.
    John K. Truss: Sets having calibre1. In: Logic Colloquium 76: Proceedings of a Conference held in Oxford in July 1976, R.O. Gandy, J.M.E. Hyland (eds.). Studies in Logic and the Foundations of Mathematics, vol. 87, pp. 595–612. North-Holland, Amsterdam (1977) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Institut für MathematikUniversität ZürichZürichSwitzerland

Personalised recommendations