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Forcings with ideals and simple forcing notions

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Abstract

Using generic ultrapower techniques we prove the following statements:

  1. (1)

    for every sequence 〈μ n |n <ω〉 of 0–1σ-additive measures over the set of reals, there exists a set which is nonmeasurable in eachμ n ,

  2. (2)

    there is no nowhere primeσ-complete ℵ0-dense ideal,

  3. (3)

    ifI is a nowhere prime ideal over a setX then add (I) ≦d(I),

  4. (4)

    suppose thatμ is aσ-additive total nowhere prime probability measure over a setX, then add (μ) <d(μ), in particular, ifμ is a real valued measure on the continuum, then the measure algebra cannot have countable density,

  5. (5)

    there is noσ-complete idealI over a setX such that the forcing withI is isomorphic to the Cohen real forcing or to the random real forcing or to the Hechler real forcing or to the Sacks real forcing.

Some general conditions on forcing preventing it for being isomorphic to the forcing with an ideal are formulated.

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References

  1. J. Baumgartner and A. Taylor,Saturation properties of ideals in generic extensions, II, Trans. Am. Math. Soc.271 (1982), 587–609.

    Article  MATH  MathSciNet  Google Scholar 

  2. P. Erdös,Some remarks on set theory, Proc. Am. Math. Soc.1 (1950), 127–141.

    Article  MATH  Google Scholar 

  3. R. Frankiewicz and A. Gutek,Some remarks on embeddings of Boolean algebras and the topological spaces I, Bull. Acad. Polon. Sci. (Math.)29 (1982), 471–476.

    MathSciNet  Google Scholar 

  4. D. Fremlin,List of problems, circulated notes.

  5. D. Fremlin,Consequences of Martin’s Axiom, Cambridge University Press, 1984.

  6. M. Gitik and S. Shelah,Cardinal preserving ideals, J. Symb. Logic, submitted.

  7. L. Grinblat,On σ-algebras of abstract sets, Isr. J. Math., to appear.

  8. S. Hechler,Classifying almost-disjoint families with applications to βN ∖ N, Isr. J. Math.10 (1971), 413–432.

    MATH  MathSciNet  Google Scholar 

  9. T. Jech,Set Theory, Academic Press, 1978.

  10. T. Jech and K. Prikry,Cofinality of the partial ordering of functions from ω 1 into ω under eventual domination, Math. Proc. Camb. Phil. Soc.95 (1984), 25–32.

    Article  MATH  MathSciNet  Google Scholar 

  11. R. Jensen and R. Solovay,Some applications of almost disjoint sets, inMathematical Logic and Foundations of Set Theory (Y. Bar-Hillel, ed.), North-Holland, Amsterdam, 1968, pp. 84–104.

    Google Scholar 

  12. S. Kanamori and M. Magidor,The evolution of large cardinal axioms in set theory, inHigher Set Theory, Lecture Notes in Math. #669, Springer-Verlag, Berlin, 1978, pp. 99–275.

    Chapter  Google Scholar 

  13. A. Krawczyk and A. Pelc,On families of σ-complete ideals, Fund. Math.CIX (1980), 155–161.

    MathSciNet  Google Scholar 

  14. K. Kunen,Set Theory, North-Holland Studies in Logic and Foundations of Math., Vol. 102 (1980).

  15. K. Kunen,Saturated ideals, J. Symb. Logic43 (1978), 65–76.

    Article  MATH  MathSciNet  Google Scholar 

  16. K. Kunen,On measure and category, inHandbook of Set Theoretic Topology (K. Kunen and J. Vaughan, eds.), North-Holland, Amsterdam, 1984.

    Google Scholar 

  17. M. Magidor,On the existence of non-regular ultrafilters and the cardinality of ultrapowers, Trans. Am. Math. Soc.249 (1979), 97–111.

    Article  MATH  MathSciNet  Google Scholar 

  18. S. Shelah,Proper forcing, Lecture Notes in Math.940, Springer-Verlag, Berlin, 1982.

    MATH  Google Scholar 

  19. S. Shelah,Iterated forcing and normal ideals on ω 1, Isr. J. Math.60 (1987), 345–380.

    MATH  Google Scholar 

  20. S. Shelah,On normal ideals and Boolean algebras, inAround Classification Theory of Models, Lecture Notes in Math.1182, Springer-Verlag, Berlin, 1986, pp. 247–260.

    Chapter  Google Scholar 

  21. R. Solovay,Real-valued measurable cardinals, inAxiomatic Set Theory (D. Scott, ed.), Proc. Symp. Pure Math.13 (I) (1971), 397–428.

    MathSciNet  Google Scholar 

  22. R. Solovay,A model of set theory in which every set of reals is Lebesgue measurable, Ann. Math.92 (1970), 1–56.

    Article  MathSciNet  Google Scholar 

  23. A. Taylor,On saturated sets of ideals and Ulam’s problem, Fund. Math.CIX (1980), 37–53.

    Google Scholar 

  24. A. Tarski,Some problems and results relevant to the foundations of set theory, inLogic, Methodology and Philosophy of Science (E. Nagel, P. Suppes and A. Tarski, eds.), Stanford, California, 1962, pp. 125–135.

    Google Scholar 

  25. J. Truss,Sets having calibre ℵ 1, inLogic Colloquium 76 (R. Gandy and M. Hyland, eds.), North-Holland, Amsterdam, 1977, pp. 595–612.

    Google Scholar 

  26. H. Woodin, ℵ1-dense ideal on ℵ1, unpublished.

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Authors and Affiliations

  1. School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel

    Moti Gitik

  2. Hebrew University of Jerusalem, Jerusalem, Israel

    Saharon Shelah

Authors
  1. Moti Gitik
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  2. Saharon Shelah
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Additional information

The second author would like to thank the US-Israel Binational Science Foundation for partial support.

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Gitik, M., Shelah, S. Forcings with ideals and simple forcing notions. Israel J. Math. 68, 129–160 (1989). https://doi.org/10.1007/BF02772658

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  • DOI: https://doi.org/10.1007/BF02772658

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