## 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 the*P*-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 the*P*-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## Preview

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## References

- [AT]U. Abraham and S. Todorcevic,
*Partition properties of ω*_{1}*compatible with CH*, Fundamenta Mathematicae**152**(1997), 165–181.zbMATHMathSciNetGoogle Scholar - [BłSh]A. Błaszczyk and S. Shelah,
*Regular subalgebras of complete Boolean algebras*, Journal of Symbolic Logic**66**(2001), 792–800.CrossRefMathSciNetzbMATHGoogle Scholar - [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 - [Jen]R. Jensen,
*Definable sets of minimal degree*, in*Mathematical Logic and Foundations of Set Theory*(Y. Bar-Hillel, ed.), North-Holland, Amsterdam, 1970, pp. 122–128.Google Scholar - [JuSh]H. Judah and S. Shelah,
*Souslin forcing*, Journal of Symbolic Logic**53**(1988), 1182–1207.MathSciNetGoogle Scholar - [KR]N. J. Kalton and J. W. Roberts,
*Uniformly exhaustive submeasures and nearly additive set functions*, Transactions of the American Mathematical Society**278**(1983), 803–816.zbMATHCrossRefMathSciNetGoogle Scholar - [Mah]D. Maharam,
*An algebraic characterization of measure algebras*, Annals of Mathematics (2)**48**(1947), 154–167.CrossRefMathSciNetGoogle Scholar - [Mau]D. Mauldin,
*The Scottish Book, Mathematics of the Scottish Cafe*, Birkhäuser, Basel, 1981.Google Scholar - [Qu1]S. Quickert,
*Forcing and the reals*, Ph.D. Thesis, University of Bonn, 2002.Google Scholar - [Qu2]S. Quickert,
*CH and the Sacks property*, Fundamenta Mathematicae**171**(2002), 93–100.zbMATHMathSciNetGoogle Scholar - [Sh1]S. Shelah,
*How special are Cohen and random forcing*, Israel Journal of Mathematics**88**(1994), 159–174.zbMATHMathSciNetCrossRefGoogle Scholar - [Sh2]S. Shelah,
*There may be no nowhere dense ultrafilters*, in*Logic 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 - [Sh3]S. Shelah,
*On what I do not understand and have something to say*, Fundamenta Mathematicae**166**(2000), 1–82.zbMATHMathSciNetGoogle Scholar - [SZ]S. Shelah and J. Zapletal,
*Embeddings of Cohen algebras*, Advances in Mathematics**126**(1997), 93–115.zbMATHCrossRefMathSciNetGoogle Scholar - [So1]S. Solecki,
*Analytic ideals and their applications*, Annals of Pure and Applied Logic**99**(1999), 51–72.zbMATHCrossRefMathSciNetGoogle Scholar - [T]S. Todorcevic,
*A dichotomy for P-ideals of countable sets*, Fundamenta Mathematicae**166**(2000), 251–267.zbMATHMathSciNetGoogle Scholar - [Vel]B. Velickovic,
*The basis problem for CCC posets*, in*Set 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 - [Ve2]B. Velickovic,
*CCC posets of perfect trees*, Compositio Mathematica**79**(1991), 279–294.zbMATHMathSciNetGoogle Scholar