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Israel Journal of Mathematics

, Volume 147, Issue 1, pp 209–220 | Cite as

CCC forcing and splitting reals

  • Boban Velickovic
Article

Abstract

The results of this paper were motivated by a problem of Prikry who asked if it is relatively consistent with the usual axioms of set theory that every nontrivial ccc forcing adds a Cohen or a random real. A natural dividing line is into weakly distributive posets and those which add an unbounded real. In this paper I show that it is relatively consistent that every nonatomic weakly distributive ccc complete Boolean algebra is a Maharam algebra, i.e. carries a continuous strictly positive submeasure. This is deduced from theP-ideal dichotomy, a statement which was first formulated by Abraham and Todorcevic [AT] and later extended by Todorcevic [T]. As an immediate consequence of this and the proof of the consistency of theP-ideal dichotomy we obtain a ZFC result which says that every absolutely ccc weakly distributive complete Boolean algebra is a Maharam algebra. Using a previous theorem of Shelah [Sh1] it also follows that a modified Prikry conjecture holds in the context of Souslin forcing notions, i.e. every nonatomic ccc Souslin forcing either adds a Cohen real or its regular open algebra is a Maharam algebra. Finally, I also show that every nonatomic Maharam algebra adds a splitting real, i.e. a set of integers which neither contains nor is disjoint from an infinite set of integers in the ground model. It follows from the result of [AT] that it is consistent relative to the consistency of ZFC alone that every nonatomic weakly distributive ccc forcing adds a splitting real.

Keywords

Ground Model Random Real Force Notion Infinite Subset Complete Boolean Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [AT]
    U. Abraham and S. Todorcevic,Partition properties of ω 1 compatible with CH, Fundamenta Mathematicae152 (1997), 165–181.zbMATHMathSciNetGoogle Scholar
  2. [BłSh]
    A. Błaszczyk and S. Shelah,Regular subalgebras of complete Boolean algebras, Journal of Symbolic Logic66 (2001), 792–800.CrossRefMathSciNetzbMATHGoogle Scholar
  3. [Jech]
    T. Jech,Set Theory, The Third Millennium Edition, revised and expanded Series, Springer Monographs in Mathematics, 3rd rev. ed., Springer, Berlin, 2003.zbMATHGoogle Scholar
  4. [Jen]
    R. Jensen,Definable sets of minimal degree, inMathematical Logic and Foundations of Set Theory (Y. Bar-Hillel, ed.), North-Holland, Amsterdam, 1970, pp. 122–128.Google Scholar
  5. [JuSh]
    H. Judah and S. Shelah,Souslin forcing, Journal of Symbolic Logic53 (1988), 1182–1207.MathSciNetGoogle Scholar
  6. [KR]
    N. J. Kalton and J. W. Roberts,Uniformly exhaustive submeasures and nearly additive set functions, Transactions of the American Mathematical Society278 (1983), 803–816.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [Mah]
    D. Maharam,An algebraic characterization of measure algebras, Annals of Mathematics (2)48 (1947), 154–167.CrossRefMathSciNetGoogle Scholar
  8. [Mau]
    D. Mauldin,The Scottish Book, Mathematics of the Scottish Cafe, Birkhäuser, Basel, 1981.Google Scholar
  9. [Qu1]
    S. Quickert,Forcing and the reals, Ph.D. Thesis, University of Bonn, 2002.Google Scholar
  10. [Qu2]
    S. Quickert,CH and the Sacks property, Fundamenta Mathematicae171 (2002), 93–100.zbMATHMathSciNetGoogle Scholar
  11. [Sh1]
    S. Shelah,How special are Cohen and random forcing, Israel Journal of Mathematics88 (1994), 159–174.zbMATHMathSciNetCrossRefGoogle Scholar
  12. [Sh2]
    S. Shelah,There may be no nowhere dense ultrafilters, inLogic Colloq. 95. Proceedings of the Annual European Summer Meeting of the ASL, Haifa, Israel, Aug. 1995 (A. Makowsky and Z. Johann, eds.), Lecture Notes in Logic, Vol. 11, Springer-Verlag, Berlin, 1998, pp. 305–324.Google Scholar
  13. [Sh3]
    S. Shelah,On what I do not understand and have something to say, Fundamenta Mathematicae166 (2000), 1–82.zbMATHMathSciNetGoogle Scholar
  14. [SZ]
    S. Shelah and J. Zapletal,Embeddings of Cohen algebras, Advances in Mathematics126 (1997), 93–115.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [So1]
    S. Solecki,Analytic ideals and their applications, Annals of Pure and Applied Logic99 (1999), 51–72.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [T]
    S. Todorcevic,A dichotomy for P-ideals of countable sets, Fundamenta Mathematicae166 (2000), 251–267.zbMATHMathSciNetGoogle Scholar
  17. [Vel]
    B. Velickovic,The basis problem for CCC posets, inSet Theory (Piscataway, NJ, 1999), DIMACS Series in Discrete Mathematics and Theoretical Computer Science 58, American Mathematical Society, Providence, RI, 2002, pp. 149–160.Google Scholar
  18. [Ve2]
    B. Velickovic,CCC posets of perfect trees, Compositio Mathematica79 (1991), 279–294.zbMATHMathSciNetGoogle Scholar

Copyright information

© The Hebrew University Magnes Press 2005

Authors and Affiliations

  • Boban Velickovic
    • 1
  1. 1.Equipe de LogiqueUniversité de Paris 7ParisFrance

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