Classification theory: 1985

  • John T. Baldwin
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1292)


In this brief survey we have tried to place classification theory in a historical setting. We have described some of the notable results in the subject and indicated some current problems. We have touched on the lively interaction between stability theory and finite group theory. We have explored in more detail the beginnings of what we hope will be an equally fruitful interaction with the theory of algebraic groups.

Our survey has not even mentioned all the papers in this volume. Who knows what the future will bring?


Algebraic Group Classification Theory Order Theory Order Property Nonabelian Simple Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Ahlbrandt G. 1984 Totally categorical theories of modular type. University of Illinois at Chicago, 1984. 108 pp. Dissertation.Google Scholar
  2. Ahlbrandt G. and Ziegler M. 1986 Quasi-finitely axiomatizable totally categorical theories. Annals of Pure and Applied Logic, vol.30 (1986), pp.63–83.MathSciNetCrossRefzbMATHGoogle Scholar
  3. Baldwin J.T. 1976 Review of Mazoyer. Mathematical Reviews, vol.52 (1976), pp.408.Google Scholar
  4. 1987 Fundamentals of Stability Theory. Springer-Verlag, 1987. 454pp.Google Scholar
  5. Baldwin J.T. and Lachlan A.H. 1971 On strongly minimal sets. The Journal of Symbolic Logic, vol.36 (1971), pp.79–96.MathSciNetCrossRefzbMATHGoogle Scholar
  6. -1973 On universal Horn classes categorical in some infinite power. Algebra Universalis, vol.3 (1973), pp.98–111.MathSciNetCrossRefzbMATHGoogle Scholar
  7. Baldwin J.T. and McKenzie R. 1982 Counting models in universal Horn classes. Algebra Universalis, vol. 15 (1982), pp.359–384.MathSciNetCrossRefzbMATHGoogle Scholar
  8. Baur W., Cherlin G.L. and Macintyre A. 1979 Totally categorical groups and rings. Journal of Algebra, vol.57 (1979), pp.407–440.MathSciNetCrossRefzbMATHGoogle Scholar
  9. Berline C. and Cherlin G. 1983 QE rings in characteristic p n. Journal of Symbolic Logic, vol.48 (1983), pp.140–162.MathSciNetCrossRefzbMATHGoogle Scholar
  10. Berline C. and Lascar D. 1986 Groupes superstables. Annals of Pure and Applied Logic, vol.30 (1986), pp.1–45.MathSciNetCrossRefzbMATHGoogle Scholar
  11. Buechler S. 1984 Geometry of weakly minimal types. Journal of Symbolic Logic, vol.50 (1984), pp.1044–1054.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 1985 Invariants for ω-categorical, ω-stable theories. Israel Journal of Mathematics, vol.52 (1985), pp.65–81.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 1986 Locally modular theories of finite rank. Annals of Pure and Applied Logic, vol.30 (1986), pp.83–95.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 1986a Maximal chains in the fundamental order. Journal of Symbolic Logic, vol.51 (1986), pp.323–326.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 1987 The classification of small weakly minimal sets I. In: Classification Theory: Chicago, 1985, ed. by Baldwin J. Springer-Verlag, 1987. Preprint.Google Scholar
  16. Cherlin G.L. 1983 Totally categorical structures. In: Proceedings of the International Congress of Mathematicians, Warsaw 1983. 1983. pp.301–306.Google Scholar
  17. 1979 Groups of small Morley rank. Annals of Mathematical Logic, vol.17 (1979), pp.1–28.MathSciNetCrossRefzbMATHGoogle Scholar
  18. Cherlin G.L., Harrington L. and Lachlan A.H. 1985 χ 0-categorical, χ0-stable structures. Annals of Pure and Applied Logic, vol.2 (1985), pp.103–135.MathSciNetCrossRefzbMATHGoogle Scholar
  19. Ehrenfeucht A. and Mostowski A.W. 1956 Models of axiomatic theories admitting automorphisms. Fundamenta Mathematicae, vol.43 (1956), pp.50–68.MathSciNetzbMATHGoogle Scholar
  20. Evans D. 198? Homogeneous Geometries. 1985. Preprint.Google Scholar
  21. Fraïssé R. 1953 Sur l'extension aux relations de quelques propriétés connues des ordres. Academie des Sciences de Paris. Comptes Rendus, vol.237 (1953), pp.508–510.zbMATHGoogle Scholar
  22. Fuchs L. 1970 Infinite Abelian Groups I. Academic Press, 1970.Google Scholar
  23. 1973 Infinite Abelian Groups II. Academic Press, 1973.Google Scholar
  24. Grossberg R. and Shelah S. 1983 On universal locally finite groups. Israel Journal of Mathematics, vol.44 (1983), pp.289–302.MathSciNetCrossRefzbMATHGoogle Scholar
  25. Gruson L. and Jensen C. 1976 Deux applications de la notion de L-dimension. Academie des Sciences de Paris. Comptes Rendus, vol.282 (1976), pp.A23–A24.MathSciNetzbMATHGoogle Scholar
  26. Grossberg R. and Shelah S. 1986 On the number of nonisomorphic models of an infinitary theory which has the infinitary order property. Journal of Symbolic Logic, vol.51 (1986).Google Scholar
  27. Hrushovski E. 1986 Contributions to stable model theory. University of California, Berkeley, 1986. ii+108pp. Dissertation.Google Scholar
  28. Hrushovski E. and Pillay A. 1986 Weakly normal groups. In: Proceedings of Logic Colloquium 85, Paris. North Holland, 1986.Google Scholar
  29. Knight J & Lachlan A.H. 1987 Shrinking, stretching, and codes for homogeneous structures. In: Classification Theory: Chicago, 1985, ed. by Baldwin J. Springer-Verlag, 1987.Google Scholar
  30. Lachlan A.H. 1972 A property of stable theories. Fundamenta Mathematicae, vol.77 (1972), pp.9–20.MathSciNetzbMATHGoogle Scholar
  31. 1974 Two conjectures regarding the stability of ω-categorical theories. Fundamenta Mathematicae, vol.81 (1974), pp.133–145.MathSciNetzbMATHGoogle Scholar
  32. Lascar D. 1986 Introduction à la stabilité. Longman, 1986. 220pp.Google Scholar
  33. 198? Les groupes ω-stable de rang fini. Preprint.Google Scholar
  34. Laskowski M.C. 198? Uncountable theories which are categorical in a higher power. 31pp. Preprint.Google Scholar
  35. Leiner F. 1982 Lokal endlich-π separierte Gruppen mit Minimalbedingung für Zentralisatoren. Archiv der Mathematik, vol.39 (1982), pp.407–413.MathSciNetCrossRefzbMATHGoogle Scholar
  36. Loveys J. 1985 Certain weakly minimal theories. Simon Fraser University, 1985. 104pp. Dissertation.Google Scholar
  37. Macintyre A. and Shelah S. 1976 Uncountable universal locally finite groups. Journal of Algebra, vol.43 (1976), pp.168–175.MathSciNetCrossRefzbMATHGoogle Scholar
  38. Makowsky J. 1985 Abstract embedding relations. In: Model Theoretic Logics, ed. by Barwise J. & Feferman S. Springer-Verlag, 1985. pp.747–792.Google Scholar
  39. Mazoyer J. 1975 Sur les théories catégoriques finitement axiomatisables (English summary). Academie des Sciences de Paris. Comptes Rendus, vol.281 (1975), pp.A403–A406.MathSciNetGoogle Scholar
  40. Morley M. 1965 Categoricity in power. Transactions of the American Mathematical Society, vol.114 (1965), pp.514–538.MathSciNetCrossRefzbMATHGoogle Scholar
  41. Morley M. and Vaught R.L. 1962 Homogeneous universal models. Mathematica Scandinavica, vol.11 (1962), pp.37–57.MathSciNetzbMATHGoogle Scholar
  42. Nesin A. 1985 Groups of finite Morley rank. Yale University, 1985. 119pp. Dissertation.Google Scholar
  43. Pillay A. 1983 An introduction to stability theory. Clarendon Press, Oxford, 1983. xi + 146pp.zbMATHGoogle Scholar
  44. 1986 Superstable groups of finite rank without pseudoplanes. Annals of Pure and Applied Logic, vol.30 (1986), pp.95–101.MathSciNetCrossRefzbMATHGoogle Scholar
  45. 1987 Simple Superstable theories. In: Classification Theory: Chicago, 1985, ed. by Baldwin J. Springer-Verlag, 1987.Google Scholar
  46. 198? Stable theories, pseudoplanes and the number of countable models. Annals of Pure and Applied Logic Preprint.Google Scholar
  47. Pillay A. and Steinhorn C. 1984 Definable sets in ordered structures. Bulletin of the American Mathematical Society, vol.11 (1984), pp.159–161.MathSciNetCrossRefzbMATHGoogle Scholar
  48. Poizat B. 1981 Sous-groupes définissables d'un groupe stable. The Journal of Symbolic Logic, vol.46 (1981), pp.137–146.MathSciNetCrossRefzbMATHGoogle Scholar
  49. 1985 Cours de théorie des modéles. Nur al-Mantiq wal-Ma'rifah 82 Rue Racine, 69100 Villeurbanne, France, 1985. 584pp.Google Scholar
  50. 198? MM. Borel, Tits, Zil'ber et le général nonsense (198?). Preprint.Google Scholar
  51. Reineke J. 1975 Minimale Gruppen. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol.21 (1975), pp.357–359.MathSciNetCrossRefzbMATHGoogle Scholar
  52. Shelah S. 1971 Stability, the f.c.p. and superstability; model theoretic properties of formulas in first-order theories. Annals of Mathematical Logic, vol.3 (1971), pp.271–362.CrossRefzbMATHGoogle Scholar
  53. 1975 The lazy model theorist's guide to stability. Logique et Analyse. Nouvelle Série, vol.18 (1975), pp.241–308.zbMATHGoogle Scholar
  54. 1978 Classification Theory and the Number of Nonisomorphic Models. North-Holland. 1978. XVI+544pp.Google Scholar
  55. -1980 Independence results. The Journal of Symbolic Logic, vol.45 (1980), pp.563–573.MathSciNetCrossRefzbMATHGoogle Scholar
  56. 1982 The spectrum problem I, ℵ -saturated models the main gap. Israel Journal of Mathematics, vol.43 (1982), pp.324–356.MathSciNetCrossRefzbMATHGoogle Scholar
  57. 1983 Classification theory for non-elementary classes I, the number of uncountable models ψ ε \(L_{\omega _1 ,\omega }\). Part A. Israel Journal of Mathematics, vol.46 (1983), pp.212–240.MathSciNetCrossRefGoogle Scholar
  58. -1985 Classification of first order theories which have a structure theory. Bulletin of the American Mathematical Society, vol.12 (1985), pp.227–232.CrossRefzbMATHGoogle Scholar
  59. 1987 Universal classes. In: Classification Theory: Chicago, 1985, ed. by Baldwin J. Springer-Verlag, 1987.Google Scholar
  60. 198? Classification theory and the number of nonisomorphic models (2nd edition). North-Holland. 198?. Preprint.Google Scholar
  61. 198?a The spectrum problem III, Universal theories. Israel Journal of Mathematics Preprint.Google Scholar
  62. Steinberg R. 1972/73 1972/73 Abstract homomorphisms of simple algebraic groups. In: Seminaire Bourbaki 25è année. 1972/73.Google Scholar
  63. Thomas S. 1983 Classification theory of simple locally finite groups. Bedford College, University of London, 1983. 89pp. Dissertation.Google Scholar
  64. Vaughn J. 1985 N 1-categorical locally modular theories. University of Illinois at Chicago, 1985. 67pp. Dissertation.Google Scholar
  65. Vaught R.L. 1961 Denumerable models of complete theories. In: Infinitistic Methods Proceedings of the Symposium on Foundations of Mathematics: Infinitistic Methods. Panstwowe Wydawnictwo Naukowe (PWN), 1961. pp.303–321.Google Scholar
  66. Ziegler M. 1984 Model theory of modules. Annals of Pure and Applied Logic, vol.26/2 (1984), pp.149–213.MathSciNetCrossRefzbMATHGoogle Scholar
  67. Zil'ber B.I. 1977 Groups and rings, the theory of which is categorical (Russian.English summary). Fundamenta Mathematicae, vol.95 (1977), pp.173–188.MathSciNetzbMATHGoogle Scholar
  68. 1981 Totally categorical structures and combinatorial geometries. (English translation). Doklady Akademii Nauk SSSR (1981), pp.149–151. Russian original: pp.1039–41.Google Scholar
  69. -1984 Some model theory of simple algebraic groups over algebraically closed fields. Colloquium Mathematicum, vol.48 (1984), pp.173–180.MathSciNetGoogle Scholar
  70. Zimmerman W. 1977 Rein injektive direkte Summen von Moduln. Communications in Algebra, vol.5 (1977), pp.1083–1117.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • John T. Baldwin
    • 1
  1. 1.University of IllinoisChicago

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