• Frank O. Wagner
Part of the Mathematics and Its Applications book series (MAIA, volume 503)


Mathematics often proceeds from the specific to the general, and the development of simplicity theory is no exception to the rule. It began with Michael Morley’s study of uncountably categorical theories, where he defined ω-stability, and for some time remained in the categorical context. This changed when Saharon Shelah embarked on an ambitious programme of classifying the models of a complete first-order theory, using his newly invented notion of “forking” and Rowbottom’s “stability”. Unstable theories have the maximal number of models and are thus considered unclassifiable; nevertheless, he tried in [149, 152] to extend the framework, defining a well-behaved class of unstable first-order theories which he called “simple unstable”. However, symmetry of forking for those theories eluded him, and at the time those papers did not receive the attention they deserved.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliographical Remarks

  1. [2]
    John T. Baldwin. aT is finite for s`il-categorical T. Transactions of the American Mathematical Society, 181: 37–51, 1973.zbMATHGoogle Scholar
  2. [6]
    John T. Baldwin and Alistair H. Lachlan. On strongly minimal sets. Journal of Symbolic Logic, 36: 79–96, 1971.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    John T. Baldwin. Fundamentals of Stability Theory. Springer-Verlag, Berlin, Germany, 1988.Google Scholar
  4. [45]
    Wilfrid Hodges. Model Theory. Cambridge University Press, Cambridge, UK, 1993.CrossRefzbMATHGoogle Scholar
  5. [125]
    Bruno P. Poizat. Cours de théorie des modèles. Nur Al-Mantiq WalMa’rifah, Villeurbanne, France, 1985.Google Scholar
  6. [106]
    Anand Pillay. Geometric Stability Theory. Oxford University Press, Oxford, UK, 1996.zbMATHGoogle Scholar
  7. [147]
    Saharon Shelah. Classification Theory. North-Holland, Amsterdam, The Netherlands, 1978.Google Scholar
  8. [16]
    Steven Buechler. Essential Stability Theory. Springer-Verlag, Berlin, Germany, 1996.Google Scholar
  9. [85]
    Daniel Lascar. Stability in Model Theory. Longman, New York, USA, 1987.Google Scholar
  10. [103]
    Anand Pillay. An Introduction to Stability Theory. Clarendon Press, Oxford, UK, 1983.zbMATHGoogle Scholar
  11. [100]
    Michael Morley. Categoricity in power. Transactions of the American Mathematical Society, 114: 514–538, 1965.CrossRefGoogle Scholar
  12. [118]
    Bruno P. Poizat. Modèles premiers d’une théorie totalement transcendante. In Théories Stables II. Institut Henri Poincaré, Paris, France, 1981.Google Scholar
  13. [119]
    Bruno P. Poizat. Sous-groupes définissables d’un groupe stable. Journal of Symbolic Logic, 46: 137–146, 1981.CrossRefzbMATHMathSciNetGoogle Scholar
  14. [120]
    Bruno P. Poizat. Théories instables. Journal of Symbolic Logic, 46: 513522, 1981.Google Scholar
  15. [121]
    Bruno P. Poizat. Groupes stables, avec types génériques réguliers. Journal of Symbolic Logic, 48: 339–355, 1983.CrossRefzbMATHMathSciNetGoogle Scholar
  16. [122]
    Bruno P. Poizat. Paires de structures stables. Journal of Symbolic Logic, 48: 239–249, 1983.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [123]
    Bruno P. Poizat. Post-scriptum à “Théories instables”. Journal of Symbolic Logic, 48: 60–62, 1983.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [124]
    Bruno P. Poizat. Une théorie de Galois imaginaire. Journal of Symbolic Logic, 48, 1983.Google Scholar
  19. [125]
    Bruno P. Poizat. Cours de théorie des modèles. Nur Al-Mantiq WalMa’rifah, Villeurbanne, France, 1985.Google Scholar
  20. [126]
    Bruno P. Poizat. Groupes Stables. Nur Al-Mantiq Wal-Ma’rifah, Villeurbanne, France, 1987.Google Scholar
  21. [127]
    Bruno P. Poizat. Corps de rang deux. In The 3rd Barcelona Logic Meeting, volume 9, pages 31–44. Centre de Recerca Matematica, Barcelona, Spain, 1997.Google Scholar
  22. [128]
    Bruno P. Poizat. Le carré de l’égalité. Journal of Symbolic Logic,to appear.Google Scholar
  23. [129]
    Bruno P. Poizat. L’égalité au cube. Journal of Symbolic Logic,to appear.Google Scholar
  24. [130]
    Massoud Pourmandian. Simple Generic Structures. PhD thesis, University of Oxford, Oxford, UK, 1999.Google Scholar
  25. [131]
    Mike Prest. Model Theory of Modules (LMS LN 130). Cambridge University Press, Cambridge, UK, 1988.CrossRefGoogle Scholar
  26. [132]
    F. P. Ramsey. On a problem of formal logic. Proceedings of the London Mathematical Society, 30: 427–489, 1930.Google Scholar
  27. [133]
    Joachim Reineke. Minimale Gruppen. Zeitschrift für Mathematische Logik, 21: 357–359, 1975.zbMATHGoogle Scholar
  28. [134]
    F. Rowbottom. The Los conjecture for uncountable theories. Notices of the American Mathematical Society, 11: 248, 1964.Google Scholar
  29. [135]
    C. Ryll-Nardzewski. On the categoricity in power 1’ o. Bulletin de l’Académie Polonaise des Sciences, 7: 545–548, 1959.zbMATHMathSciNetGoogle Scholar
  30. [136]
    G. Schlichting. Operationen mit periodischen Stabilisatoren. Archiv der Mathematik (Basel), 34: 97–99, 1980.zbMATHMathSciNetGoogle Scholar
  31. [139]
    Ziv Shami. A natural finite equivalence relation definable in low theories and a criterion for Lstp=stp. Preprint, 1997.Google Scholar
  32. [140]
    Saharon Shelah. Categoricity of Classes of Models. PhD thesis, The Hebrew University, Jerusalem, Israel, 1969.Google Scholar
  33. [141]
    Saharon Shelah. Stable theories. Israel Journal of Mathematics, 7: 187202, 1969.Google Scholar
  34. [142]
    Saharon Shelah. Every two elementarily equivalent models have isomorphic ultrapowers. Israel Journal of Mathematics, 10: 224–233, 1971.CrossRefzbMATHMathSciNetGoogle Scholar
  35. [143]
    Saharon Shelah. Stability, the f.c.p., and superstability; model-theoretic properties of formulas in first-order theories. Annals of Mathematical Logic, 3: 271–362, 1971.CrossRefzbMATHGoogle Scholar
  36. [144]
    Saharon Shelah. Uniqueness and characterization of prime models over sets for totally transcendental first-order theories. Journal of Symbolic Logic, 37: 107–113, 1972.CrossRefzbMATHMathSciNetGoogle Scholar
  37. [148]
    Saharon Shelah. On uniqueness of prime models. Journal of Symbolic Logic, 44: 215–220, 1979.CrossRefzbMATHMathSciNetGoogle Scholar
  38. [8]
    Andreas Baudisch. Decidability and stability of free nilpotent Lie algebras and free nilpotent p-groups of finite exponent. Annals of Mathematical Logic, 23: 1–25, 1982.CrossRefzbMATHMathSciNetGoogle Scholar
  39. [23]
    Olivier Chapuis. Universal theory of certain solvable groups and bounded Ore group-rings. Journal of Algebra, 176: 368–391, 1995.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Frank O. Wagner
    • 1
  1. 1.Institut Girard DesarguesUniversité Claude Bernard (Lyon-1)VilleurbanneFrance

Personalised recommendations