Israel Journal of Mathematics

, 173:1 | Cite as

Dependent first order theories, continued



A dependent theory is a (first order complete theory) T which does not have the independence property. A major result here is: if we expand a model of T by the traces on it of sets definable in a bigger model then we preserve its being dependent. Another one justifies the cofinality restriction in the theorem (from a previous work) saying that pairwise perpendicular indiscernible sequences, can have arbitrary dual-cofinalities in some models containing them. We introduce “strongly dependent” and look at definable groups; and also at dividing, forking and relatives.


  1. [BaPo98]
    Y. Baisalov and B. Poizat, Paires de structures o-minimales, Journal of Symbolic Logic 63 (1998), 570–578.MATHCrossRefMathSciNetGoogle Scholar
  2. [BBSh:815]
    B. Baizhanov, J. Baldwin and S. Shelah, Subsets of superstable structures are weakly benign, Journal of Symbolic Logic 70 (2005), 142–150, math.LO/0303324.MATHCrossRefMathSciNetGoogle Scholar
  3. [BlBn00]
    J. T. Baldwin and M. Benedikt, Stability theory, permutations of indiscernibles, and embedded finite model theory, Transactions of the American Mathematical Society 352 (2000), 4937–4969.MATHCrossRefMathSciNetGoogle Scholar
  4. [CoSh:919]
    M. Cohen and S. Shelah, Stable theories and representation over sets, preprint.Google Scholar
  5. [Cmf77]
    W. W. Comfort, Ultrafilters: some old and some new results, Bulletin of the American Mathematical Society 83 (1977), 417–455.MATHCrossRefMathSciNetGoogle Scholar
  6. [FiSh:E50]
    E. Firstenberg and S. Shelah, Pseudo unidimensional dependent theories. For more information please contact the author.Google Scholar
  7. [Ke87]
    J. H. Keisler, Measures and forking, Annals of Pure and Applied Logic 34 (1987),119–169.MATHCrossRefMathSciNetGoogle Scholar
  8. [Po81]
    B. Poizat, Théories instables, Journal of Symbolic Logic 46 (1981), 513–522.MATHCrossRefMathSciNetGoogle Scholar
  9. [Sh:3]
    S. Shelah, Finite diagrams stable in power, Annals of Mathematical Logic 2 (1970), 69–118.MATHCrossRefGoogle Scholar
  10. [Sh:93]
    S. Shelah, Simple unstable theories, Annals of Mathematical Logic 19 (1980), 177–203.MATHCrossRefMathSciNetGoogle Scholar
  11. [Sh:c]
    S. Shelah, Classification theory and the number of nonisomorphic models, in Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam, xxxiv+705 pp., 1990.Google Scholar
  12. [Sh:715]
    S. Shelah, Classification theory for elementary classes with the dependence property — a modest beginning, Scientiae Mathematicae Japonicae 59,No. 2; (special issue: e9, 503–544) (2004), 265–316, math.LO/0009056.MATHMathSciNetGoogle Scholar
  13. [Sh:876]
    S. Shelah, Minimal bounded index subgroup for dependent theories, Proceedings of the American Mathematical Society 136 (2008), 1087–1091.MATHCrossRefMathSciNetGoogle Scholar
  14. [Sh:839]
    S. Shelah, Stable frames and weight.Google Scholar
  15. [Sh:863]
    S. Shelah, Strongly dependent theories, submitted.Google Scholar
  16. [Sh:877]
    S. Shelah, Dependent T and Existence of limit models, preprint, math. LO/0609636.Google Scholar
  17. [Sh:886]
    S. Shelah, Definable groups for dependent and 2-dependent theories, submitted.Google Scholar
  18. [Sh:900]
    S. Shelah, Dependent theories and the generic pair conjecture, Communications in Contemporary Mathematics, submitted, math.LO/0702292Google Scholar
  19. [Sh:950]
    S. Shelah, A dependent dream and recounting types, preprint.Google Scholar
  20. [Sh:F660]
    S. Shelah, Toward the main gap by fatness, a modest beginning.Google Scholar
  21. [Sh:F705]
    S. Shelah, Representation over orders of elementary classes.Google Scholar
  22. [Sh:F906]
    S. Shelah, On incompctness in singulars (for Ext, consistently incompact).Google Scholar

Copyright information

© Hebrew University Magnes Press 2009

Authors and Affiliations

  1. 1.Einstein Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael
  2. 2.Department of Mathematics, Hill Center-Busch Campus RutgersThe State University of New JerseyPiscatawayUSA

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