Proper Forcing



The aim of this chapter is to develop the theory of proper forcings and their iteration and to provide interesting examples of its usefulness and range of applications. Our presentation is detailed and should be accessible to any reader who is familiar with the Solovay and Tennebaum technique of finite support iteration and the proof that the c.c.c. property is preserved. We present the basic preservation theorem of the properness property under countable support iteration, and continue with additional preservation theorems—all due to Shelah. Each abstract preservation theorem is followed by an application which shows its relevance. A longer section deals with posets that add no new countable sets and their iteration.


Chromatic Number Continuum Hypothesis Countable Support Elementary Substructure Supercompact Cardinal 
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  1. [1]
    James E. Baumgartner. Iterated forcing. In Adrian R.D. Mathias, editor, Surveys in Set Theory, volume 87 of London Mathematical Society Lecture Note Series, pages 1–59. Cambridge University Press, London, 1983. Google Scholar
  2. [2]
    W. Wistar Comfort and Stylianos Negrepontis. The Theory of Ultrafilters. Springer, Berlin, 1974. MATHGoogle Scholar
  3. [3]
    Keith J. Devlin and Håvard Johnsbråten. The Souslin Problem, volume 405 of Lecture Notes in Mathematics. Springer, Berlin, 1974. MATHGoogle Scholar
  4. [4]
    Martin Goldstern. Tools for your forcing constructions. In H. Judah, editor, Set Theory of the Reals, volume 6 of Israel Mathematical Conference Proceedings, pages 305–360, 1992. Google Scholar
  5. [5]
    András Hajnal and Attila Máté. Set mappings, partitions, and chromatic numbers. In H. Rose and J. Shepherdson, editors, Logic Colloquium ’73, pages 347–379. North-Holland, Amsterdam, 1975. CrossRefGoogle Scholar
  6. [6]
    Stephen Hechler. On the existence of certain cofinal subsets of ω ω. In Thomas J. Jech, editor, Axiomatic Set Theory II, volume 13, Part II of Proceedings of Symposia in Pure Mathematics, pages 155–173. American Mathematical Society, Providence, 1974. Google Scholar
  7. [7]
    Thomas J. Jech. Set Theory. Springer, Berlin, 2002. Third millennium edition. Google Scholar
  8. [8]
    Ronald B. Jensen and Håvard Johnsbråten. A new construction of a non-constructible \({\Delta}^{1}_{3}\) subset of ω. Fundamenta Mathematicae, 81:279–290, 1974. MATHMathSciNetGoogle Scholar
  9. [9]
    H. Jerome Keisler. Ultraproducts and saturated models. Indagationes Mathematicae, 16:178–186, 1964. MathSciNetGoogle Scholar
  10. [10]
    Richard Laver. On the consistency of Borel’s conjecture. Acta Mathematica, 137:151–169, 1976. CrossRefMathSciNetGoogle Scholar
  11. [11]
    Adrian R.D. Mathias. Happy families. Annals of Mathematical Logic, 12:59–111, 1977. MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Saharon Shelah. Every two elementary equivalent models have isomorphic ultrapowers. Israel Journal of Mathematics, 10:224–233, 1971. MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Saharon Shelah. On cardinal invariants of the continuum. In James E. Baumgartner, Donald A. Martin, and Saharon Shelah, editors, Axiomatic Set Theory, volume 31 of Contemporary Mathematics, pages 184–207. American Mathematical Society, Providence, 1984. Google Scholar
  14. [14]
    Saharon Shelah. Vive la différence I: Nonisomorphism of ultrapowers of countable models. In Set Theory of the Continuum, volume 26 of Mathematical Sciences Research Institute Publications, pages 354–405. Springer, Berlin, 1992. Google Scholar
  15. [15]
    Saharon Shelah. Proper and Improper Forcing. Springer, Berlin, 1998. Second edition. MATHGoogle Scholar

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© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of MathematicsBen Gurion University of the NegevBeer ShevaIsrael

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