Israel Journal of Mathematics

, Volume 43, Issue 3, pp 185–224 | Cite as

S-forcing, I. A “black-box” theorem for morasses, with applications to super-Souslin trees

  • S. Shelah
  • L. Stanley
Article
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Abstract

We formulate, for regular μ>ω, a “forcing principle” Sμ which we show is equivalent to the existence of morasses, thus providing a new and systematic method for obtaining applications of morasses. Various examples are given, notably that for infinitek, if 2 k =k + and there exists a (k +, 1)-morass, then there exists ak ++-super-Souslin tree: a normalk ++ tree characterized by a highly absolute “positive” property, and which has ak ++-Souslin subtree. As a consequence we show that CH+SHℵ 2⟹ℵ2 is (inaccessible)L.

Keywords

Initial Segment Restriction Property Amalgamation Property Elementary Embedding Infinite Cardinal 
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References

  1. 1.
    J. P. Burgess,Infinitary Languages and Descriptive Set Theory, Ph.D. Thesis, University of California, Berkeley, 1974.Google Scholar
  2. 2.
    J. P. Burgess,Consistency proofs in model theory, a contribution to Jensenlehre, Ann. Math. Logic14 (1978), 1–12.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    K. Devlin,Aspects of Constructibility, Lecture Notes in Math.354, Springer-Verlag, Berlin, 1973.MATHGoogle Scholar
  4. 4.
    K. Devlin, to appear.Google Scholar
  5. 5.
    J. Gregory,Higher Souslin trees, J. Symb. Logic41 (1976), 663–671.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    T. Jech,A game theoretic property of Boolean algebras, inLogic Colloquium '77 (A. MacIntyre, L. Pacholski and J. Paris, eds.), North Holland, 1978, pp. 135–144.Google Scholar
  7. 7.
    R. B. Jensen,The fine structure of the constructible hierarchy, Ann. Math. Logic4 (1972), 229–308.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    R. Laver,Making the supercompactness of k indestructible under k-directed closed forcing, Isr. J. Math.29 (1978), 385–388.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    R. Laver and S. Shelah,The ℵ 2-Souslin-Hypothesis Trans. Am. Math. Soc.264 (1981), 411–418.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    S. Shelah,Models with second-order properties IV:A general method and eliminating diamonds, to appear.Google Scholar
  11. 11.
    S. Shelah and L. Stanley,Generalized Martin's Axiom and Souslin's Hypothesis for higher cardinals, Isr. J. Math.43 (1982), 225–236.MATHMathSciNetGoogle Scholar
  12. 12.
    L. Stanley,A Short Course on Gap-1 Morasses with a Review of the Fine Structure of L, Proceedings of the Cambridge Summer School in Set Theory, to appear.Google Scholar
  13. 13.
    L. Stanley,Large cardinals and morasses, to appear.Google Scholar
  14. 14.
    D. Velleman, Ph.D. Thesis, University of Wisconsin, Madison, 1980.Google Scholar

Copyright information

© Hebrew University 1982

Authors and Affiliations

  • S. Shelah
    • 1
    • 2
  • L. Stanley
    • 1
    • 2
  1. 1.Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael
  2. 2.Maths Pures-Les CezeauxUniversité de Clermont IIAubiereFrance

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