Model Theory

  • Angus Macintyre
Part of the Synthese Library book series (SYLI, volume 149)


I have been invited to talk in Section 1 (Pure Logic), and not in Section 2, (The Interplay between Logic and Mathematics). Since model theory is surely the scene of most interplay between logic and mathematics, and since I am uncertain as to what pure model theory is or should be, I have had problems in delineating an appropriate subject matter.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    R. Anderson, ‘A Nonstandard Representation for Brownian Motion and Itô Integration’, Israel Journal 25 (1976), 9–14.Google Scholar
  2. [2]
    J. Ax, ‘The Elementary Theory of Finite Fields’, Ann. Math. 88 (1968), 239–271.CrossRefGoogle Scholar
  3. [3]
    J. T. Baldwin and A. H. Lachlan, ‘On Strongly Minimal Sets’, J. Symbolic Logic 36 (1971), 79–96.CrossRefGoogle Scholar
  4. [4]
    J. Barwise and A. Robinson, ‘Completing Theories by Forcing’, Ann. Math. Logic 2 (1970), 119–142.CrossRefGoogle Scholar
  5. [5]
    J. Barwise, M. Kaufmann, and M. Makkai, ‘Stationary Logic’, to appear.Google Scholar
  6. [6]
    J. Barwise and J. Schlipf, ‘On Recursively Saturated Models of Arithmetic’, in Lecture Notes in Mathematics 498, Springer-Verlag, 1975, pp. 42–55.Google Scholar
  7. [7]
    W. Bauer, G. Cherlin, and A. Macintyre, Totally Categorical Groups and Rings’, submitted to J. Algebra.Google Scholar
  8. [8]
    L. Blum, Ph.D. Thesis, M.I.T., 1968.Google Scholar
  9. [9]
    S. Brown, Ph.D. Thesis, Princeton, 1976.Google Scholar
  10. [10]
    C.C. Chang, ‘Model Theory 1945-1971’, in Proceedings of the Tarski Symposium, L. Henkin et al. (eds.), A.M.S. Symposia Proceedings XXV, Providence, 1974, pp. 173–186.Google Scholar
  11. [11]
    P.J. Cohen, Set Theory and the Continuum Hypothesis, Benjamin, New York, 1966.Google Scholar
  12. [12]
    A. Ehrenfeucht and A. Mostowski, ‘Models of Axiomatic Theories Admitting Automorphisms’, Fund. Math. 43 (1956), 50–68.Google Scholar
  13. [13]
    S. Feferman and R. L. Vaught, ‘The First-Order Properties of Products of Algebraic Systems’, Fund. Math. 47 (1959), 57–103.Google Scholar
  14. [14]
    A. Friedman, Generalized Functions and Partial Differential Equations, Prentice Hall, Englewood, N.J., 1963.Google Scholar
  15. [15]
    H. Gaifman, ‘Uniform Extension Operators for Models’, in Sets, Models and Recursion Theory, Crossley (ed.), North-Holland, Amsterdam, 1967, pp. 122–155.CrossRefGoogle Scholar
  16. [16]
    F. Galvin, ‘Horn Sentences’, Ann. Math. Logic 1 (1970), 389–422.CrossRefGoogle Scholar
  17. [17]
    K. Godel, ‘Über Formal unentscheidbare Sätze der Principia Mathematica und ver-wandter Systeme. I’, Monatsh. Phys. 38 (1931), 173–198.CrossRefGoogle Scholar
  18. [18]
    A. Grothendieck, ‘Eléments de Géometrie Algébrique’, Publ. Math. I.H.E.S. 28 (1966), Chapter IV, Part III.Google Scholar
  19. [19]
    L. Henkin, ‘The Completeness of the First-Order Functional Calculus’, J. Symbolic Logic 14 (1949), 159–166.CrossRefGoogle Scholar
  20. [20]
    J. Hirschfeld and W. H. Wheeler, Forcing, Arithmetic, Division Rings, Lecture Notes in Mathematics, Springer-Verlag, 1975, p. 454.Google Scholar
  21. [21]
    L. Hörmander, Linear Partial Differential Operators, Academic Press, N.Y., 1963.Google Scholar
  22. [22]
    N. Jacobson, Lectures on Abstract Algebra, Vol. Ill, Van Nostrand, p. 1.Google Scholar
  23. [23]
    B. Jonsson, ‘Homogeneous Universal Relational Systems’, Math. Scand. 8 (1960), 137–142.Google Scholar
  24. [24]
    H. J. Keisler, Model Theory for In finitary Logic, North-Holland, Amsterdam, 1971.Google Scholar
  25. [25]
    H. J. Keisler, Starfinite Models, preprint, Madison, 1977.Google Scholar
  26. [26]
    G. Kreisel, ‘What Have We Learnt from Hilbert’s Second Problem?’, in Am. Math. Soc. Symposia Proceeding XVIII ( Hilbert Problems ), Providence, 1976, pp. 93–130.Google Scholar
  27. [27]
    G. Kreisel, ‘A Survey of Proof Theory’, J. Symbolic Logic 33 (1968), 321–388.CrossRefGoogle Scholar
  28. [28]
    J. L. Krivine, ‘Sous-espaces de dimension flnie des espaces de Banach réticulés’, Ann. Math. 104 (1976), 1–29.CrossRefGoogle Scholar
  29. [29]
    G. Kreisel and J. L. Krivine, Elements of Mathematical Logic, Model Theory, North-Holland, Amsterdam, 1967.Google Scholar
  30. [30]
    S. Lang, Introduction to Algebraic Geometry, Interscience, New York - London, 1958.Google Scholar
  31. [31]
    P. Loeb, ‘Applications of Nonstandard Analysis to Ideal Boundaries in Potential Theory’, Israel J. 25 (1976), 154–188.CrossRefGoogle Scholar
  32. [32]
    J. Łos, ‘On the Categoricity in Power of Elementary Deductive Systems and Some Related Problems’, Colloq. Math. 3 (1954), 58–62.Google Scholar
  33. [33]
    M. Morley and R. L. Vaught, ‘Homogeneous Universal Models’, Math. Scand. 11 (1962), 37–57.Google Scholar
  34. [34]
    A. Macintyre, ‘Abraham Robinson, 1918–1974’, Bull. Am. Math. Soc. 83 (1977), 646–665.CrossRefGoogle Scholar
  35. [35]
    A. Macintyre, ‘Model Completeness’, to appear in A Handbook of Mathematical Logic, J. Barwise (ed.), North-Holland, Amsterdam, 1977.Google Scholar
  36. [36]
    Ju. V. Matejasevic, ‘Recursively Enumerable Sets are Diophantine’, Dokl. Akad. Nauk SSSR 191 (1970), 279-282 (Russian).Google Scholar
  37. [37]
    F. Miraglia, Ph.D. Thesis, Yale, 1977.Google Scholar
  38. [38]
    M. Morley, ‘Categoricity in Power’, Trans. Am. Math. Soc. 114 (1965), 514 - 538.CrossRefGoogle Scholar
  39. [39]
    L. Monk, Ph.D. Thesis, Berkeley, 1975.Google Scholar
  40. [40]
    A. Mostowski, Constructive Sets with Applications, North-Holland, Amsterdam.Google Scholar
  41. [41]
    J. Paris and L. Harrington, ‘A Mathematical Incompleteness in Peano Arithmetic’, to appear in A Handbook of Mathematical Logic, J. Barwise (ed.), North-Holland, Amsterdam, 1977.Google Scholar
  42. [42]
    G. Reyes, ‘Local Definability Theory’, Ann. Math. Logic 1 (1970), 15 - 137.CrossRefGoogle Scholar
  43. [43]
    G. Reyes, Ph.D. Thesis, Berkeley, 1967.Google Scholar
  44. [44]
    A. Robinson, Introduction to Model Theory and to the Metamathematics of Algebra, 2nd ed., North-Holland, Amsterdam, 1965.Google Scholar
  45. [45]
    A. Robinson, Nonstandard Analysis, North-Holland, Amsterdam, 1966.Google Scholar
  46. [46]
    A. Robinson, ‘Topics in Nonstandard Algebraic Number Theory’, in Applications of Model Theory to Algebra, Analysis and Probability, Luxemburg (ed.), New York, 1969, pp. 1 - 17.Google Scholar
  47. [47]
    A. Robinson and P. Roquette, ‘On the Finiteness Theorem of Siegel and Mahler Concerning Diophantine Equations’, J. Number Theory 7 (1975), 121 - 176.CrossRefGoogle Scholar
  48. [48]
    C. Ryll-Nardzewski, On the Categoricity in Power א0 Bull Acad. Polon. Sci. Ser. Sci. Math. Asst. Phys. 7 (1959), 545–548.Google Scholar
  49. [49]
    G. Sacks, Saturated Model Theory, Benjamin, 1972.Google Scholar
  50. [50]
    A. Seidenberg, ‘An Elimination Theory for Differential Algebra’, Univ. California Publications in Math. 3 (1956), 31–66.Google Scholar
  51. [51]
    C. C. Chang and H. H. Keisler, Model Theory, North-Holland, 1973.Google Scholar
  52. [52]
    S. Shelah, ‘The Lazy Model-Theoretician’s Guide to Stability’, Logique et Analyse 72–72 (1975), 241–308.Google Scholar
  53. [53]
    J. Silver, ‘Some Applications of Model Theory in Set Theory’, Ann. Math. Logic 3 (1971), 45–110.CrossRefGoogle Scholar
  54. [54]
    A. Tarski and J. C. C. McKinsey, A Decision Method for Elementary Algebra and Geometry, 1st ed., The Rand Corp., Santa Monica, Calif., 1948; 2nd ed., Berkeley - Los Angeles, 1951.Google Scholar
  55. [55]
    G. Takeuti, Boolean Valued Analysis I, II, III, Preprints, Urbana, 1975.Google Scholar
  56. [56]
    R. L. Vaught, ‘Model Theory Before 1945’, pages 153–172 in Proceedings of the Tarski Symposium, L. Henkin et al. (eds.), Am. Math. Soc. Symposia Proceedings XXV, Providence, 1974.Google Scholar
  57. [57]
    R. L. Vaught, ‘Invariant Sets in Topology and Logic’, Fund. Math. 82 (1974), 269–293.Google Scholar
  58. [58]
    R. L. Vaught, ‘Denumerable Models of Complete Theories’, in Infinitistic Methods, Oxford — Warsaw, 1961, pp. 303–321.Google Scholar
  59. [59]
    S. Kochen, ‘Ultraproducts in the Theory of Models’, Am. Math. 2,74 (1961), 221–261.Google Scholar
  60. [60]
    L. van den Dries and P. Ribenborn, ‘Lefschetz Principle in Galois Theory’, Queen’s Mathematics Preprint, 1976, No. 5.Google Scholar

Copyright information

© D. Reidel Publishing Company 1981

Authors and Affiliations

  • Angus Macintyre
    • 1
  1. 1.Yale UniversityUSA

Personalised recommendations