Model Theory in Algebra, Analysis and Arithmetic: A Preface

  • Dugald Macpherson
  • Carlo Toffalori
Part of the Lecture Notes in Mathematics book series (LNM, volume 2111)


Model theory is a branch of mathematical logic dealing with mathematical structures (models) from the point of view of first order logical definability. Although comparatively young, it is now well established, its major textbooks including [6, 17, 34, 43, 53]. A typical goal of model theory is to build, study and classify mathematical universes in which some given axioms (usually expressed in a first order way) are satisfied. Thus model theory has remote roots in the birth of non-Euclidean geometries and in the effort to realize them in suitable mathematical settings where their revolutionary (non-standard) assumptions are obeyed. Some major achievements of mathematical logic in the first half of the twentieth century, such as the Gödel Compactness Theorem around 1930, underpin modern model theory.


Model Theory Independence Property Quantifier Elimination Definable Subset Exponential Field 
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.



We thank the four authors of the articles in this volume for their contributions. We also warmly thank

• the CIME director and vice-director, Pietro Zecca and Elvira Mascolo, and the whole CIME staff for the excellent organization of the course,

• all the speakers, and presenters of posters,

• all the participants (around 70, mostly young researchers),

• the Italian FIR New trends in model theory of exponentiation and its coordinator Sonia L’Innocente for their support.

We also thank the CIME Scientific Committee for accepting the proposal of this course. We hope that model theory may soon be the topic of other CIME meetings.


  1. 1.
    I.M. Aschenbrenner, A. Dolich, D. Haskell, D. Macpherson, S. Starchenko, Vapnik-Chervonenkis density in some theories without the independence property I. arXiv:1109.5438, 2011Google Scholar
  2. 2.
    J. Baldwin, Categoricity. University Lecture Series, vol. 50 (American Mathematical Society, Providence, 2009)Google Scholar
  3. 3.
    J.T. Baldwin, A.H. Lachlan, On strongly minimal sets. J. Symb. Log. 36, 79–96 (1971)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    A. Berarducci, M. Otero, Y. Peterzil, A. Pillay, A descending chain condition for groups definable in o-minimal structures. Ann. Pure Appl. Log. 134, 303–313 (2005)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    E. Bouscaren (ed.), Model Theory and Algebraic Geometry. Lecture Notes in Mathematics, vol. 1696 (Springer, New York, 1998)Google Scholar
  6. 6.
    C.C. Chang, H.J. Keisler, Model Theory (North Holland, Amsterdam, 1990)MATHGoogle Scholar
  7. 7.
    Z. Chatzidakis, D. Macpherson, A. Pillay, A. Wilkie, Model Theory with Applications to Algebra and Analysis I. London Mathematical Society Lecture Note Series, vol. 349 (Cambridge University Press, Cambridge, 2008)Google Scholar
  8. 8.
    Z. Chatzidakis, D. Macpherson, A. Pillay, A. Wilkie, Model Theory with Applications to Algebra and Analysis II. London Mathematical Society Lecture Note Series, vol. 350 (Cambridge University Press, Cambridge, 2008)Google Scholar
  9. 9.
    A. Chernikov, Theories without the tree property of the second kind. Ann. Pure Appl. Log. 165, 695–723 (2014)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    R. Cluckers, L. Lipshitz, Fields with analytic structure. J. Eur. Math. Soc. 13(4), 1147–1223 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    J. Denef, The rationality of the Poincaré series associated to the p-adic points on a variety. Invent. Math. 77, 1–23 (1984)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    J. Denef, L. van den Dries, p-adic and real subanalytic sets. Ann. Math. (2) 128, 79–138 (1988)Google Scholar
  13. 13.
    J. Denef, L. Lipshitz, T. Pheidas, J. Van Geel, Hilbert’s Tenth Problem: Relations with Arithmetic and Algebraic Geometry. Contemporary Mathematics, vol. 270 (American Mathematical Society, Providence, 2000)Google Scholar
  14. 14.
    F.J. Grunewald, D. Segal, G.C. Smith, Subgroups of finite index in nilpotent groups. Invent. Math. 93, 185–223 (1988)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    D. Haskell, A. Pillay, C. Steinhorn (eds.), Model Theory, Algebra and Geometry. MSRI Publications (Cambridge University Press, Cambridge, 2000)Google Scholar
  16. 16.
    D. Haskell, E. Hrushovski, D. Macpherson, Stable Domination and Independence in Algebraically Closed Valued Fields. ASL Lecture Notes in Logic (Cambridge University Press, Cambridge, 2007)Google Scholar
  17. 17.
    W. Hodges, Model Theory (Cambridge University Press, Cambridge, 1993)CrossRefMATHGoogle Scholar
  18. 18.
    E. Hrushovski, A new strongly minimal set. Ann. Pure Appl. Log. 62, 147–166 (1993)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    E. Hrushovski, The Mordell-Lang conjecture for function fields. J. Am. Math. Soc. 9, 667–690 (1996)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    E. Hrushovski, D. Kazhdan, Integration in valued fields, in Progress in Mathematics 850: Algebraic Geometry and Number Theory, in Honour of Vladimir Drinfeld’s 50th Birthday (Birkhäuser, Basel, 2006)Google Scholar
  21. 21.
    E. Hrushovski, F. Loeser, Non-archimedean tame topology and stably dominated types. Princeton Monograph Series (to appear)Google Scholar
  22. 22.
    E. Hrushovski, B. Martin, S. Rideau, with an appendix by R. Cluckers, Definable equivalence relations and zeta functions of groups, Math.LO/0701011v02, 2014Google Scholar
  23. 23.
    E. Hrushovski, A. Pillay, On NIP and invariant measures. J. Eur. Math. Soc. 13, 1005–1061 (2011)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    E. Hrushovski, B. Zilber, Zariski geometries. J. Am. Math. Soc. 9, 1–56 (1996)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    E. Hrushovski, Y. Peterzil, A. Pillay, Groups, measures and the NIP. J. Am. Math. Soc. 21, 563–596 (2008)Google Scholar
  26. 26.
    E. Hrushovski, A. Pillay, P. Simon, Generically stable and smooth measures in NIP theories. Trans. Am. Math. Soc. 365, 2341–2366 (2013)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    J. Koenigsmann, Defining \(\mathbb{Z}\) in \(\mathbb{Q}\), in Arithmetic of Fields, Oberwolfach Reports (2009)Google Scholar
  28. 28.
    M.C. Laskowski, Vapnik-Chervonenkis classes of definable sets. J. London Math. Soc. (2) 45, 377–384 (1992)Google Scholar
  29. 29.
    F. Loeser, Seattle lectures on motivic integration., 2008
  30. 30.
    A. Macintyre, On definable subsets of p-adic fields. J. Symb. Log. 41, 605–610 (1976)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    A. Macintyre, Model completeness, in Handbook of Mathematical Logic, ed. by J. Barwise (North Holland, Amsterdam, 1977), pp. 139–180CrossRefGoogle Scholar
  32. 32.
    A. Macintyre (ed.), Connections between Model Theory and Algebraic and Analytic Geometry (Quaderni di Matematica, Aracne, 2000)MATHGoogle Scholar
  33. 33.
    A. Macintyre, A. Wilkie, On the decidability of the real exponential field, in Kreiseliana (A. K. Peters, Wellesley, 1996), pp. 441–467Google Scholar
  34. 34.
    D. Marker, Model Theory: An Introduction. Graduate Texts in Mathematics, vol. 217 (Springer, New York, 2002)Google Scholar
  35. 35.
    Y. Matiyasevich, Hilbert’s Tenth Problem. Foundations of Computing Series (MIT Press, Cambridge, 1993)Google Scholar
  36. 36.
    B. Mazur, The topology of rational points. Exp. Math. 1, 35–45 (1992)MathSciNetMATHGoogle Scholar
  37. 37.
    B. Mazur, K. Rubin, Ranks of twists of elliptic curves and Hilbert’s Tenth Problem. Invent. Math. 181, 541–575 (2010)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    M. Morley, Categoricity in power. Trans. Am. Math. Soc. 114, 514–538 (1965)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    T. Pheidas, K. Zahidi, Decision problems in algebra and analogues of Hilbert’s tenth problem, in Model Theory with Applications to Algebra and Analysis II. London Mathematical Society Lecture Note Series, vol. 350 (Cambridge University Press, Cambridge, 2008)Google Scholar
  40. 40.
    J. Pila, O-minimality and the André-Oort conjecture for \(\mathbb{C}^{n}\). Ann. Math. (2) 173, 1779–1840 (2011)Google Scholar
  41. 41.
    J. Pila, A. Wilkie, The rational points of a definable set. Duke Math. J. 133, 591–616 (2006)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    A. Pillay, Geometric Stability Theory (Oxford University Press, Oxford, 1996)MATHGoogle Scholar
  43. 43.
    B. Poizat, A Course in Model Theory: An Introduction to Contemporary Mathematical Logic (Springer, New York, 2000)CrossRefGoogle Scholar
  44. 44.
    B. Poonen, Undecidability in number theory. Not. Am. Math. Soc. 55, 344–350 (2008)MathSciNetMATHGoogle Scholar
  45. 45.
    B. Poonen, Hilbert’s Tenth problem over rings of number-theoretic interest (2010).
  46. 46.
    A. Robinson, Complete Theories, Studies in Logic (North-Holland, Amsterdam, 1956)Google Scholar
  47. 47.
    W. Schmid, K. Vilonen, Characteristic cycles of constructible sheaves. Invent. Math. 124, 451–502 (1996)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    S. Shelah, Classification Theory (North Holland, Amsterdam, 1990)MATHGoogle Scholar
  49. 49.
    S. Shelah, Dependent first order theories, continued. Isr. J. Math. 173, 1–60 (2009)CrossRefMATHGoogle Scholar
  50. 50.
    A. Shlapentokh, Hilbert’s Tenth Problem. Diophantine Classes and Other Extensions to Global Fields. New Mathematical Monographs, vol. 7 (Cambridge University Press, Cambridge, 2007)Google Scholar
  51. 51.
    A. Tarski, Contributions to the theory of models I. Indag. Math. 16, 572–581 (1954)Google Scholar
  52. 52.
    A. Tarski, Contributions to the theory of models II. Indag. Math. 16, 582–588 (1954)Google Scholar
  53. 53.
    K. Tent, M. Ziegler, A Course in Model Theory. Lecture Notes in Logic, vol. 40 (Cambridge University Press, Cambridge, 2012)Google Scholar
  54. 54.
    L. van den Dries, Alfred Tarki’s elimination theory for reals closed fields. J. Symb. Log. 53, 7–19 (1988)CrossRefMATHGoogle Scholar
  55. 55.
    L. van den Dries, Tame Topology and o-minimal Structures. London Mathematical Society Lecture Notes Series, vol. 248 (Cambridge University Press, Cambridge, 1998)Google Scholar
  56. 56.
    F. Wagner, Simple Theories (Kluwer, Dordrecht, 2000)CrossRefMATHGoogle Scholar
  57. 57.
    A. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Am. Math. Soc. 9, 1051–1094 (1996)MathSciNetCrossRefMATHGoogle Scholar
  58. 58.
    B. Zilber, The structure of models of uncountably categorical theories, in ICM-Varsavia 1983 (North Holland, Amsterdam, 1984), pp. 359–368Google Scholar
  59. 59.
    B. Zilber, Pseudoexponentiation on algebraically closed fields of characteristic 0. Ann. Pure Appl. Log. 132, 67–95 (2004)MathSciNetCrossRefGoogle Scholar
  60. 60.
    B. Zilber, Zariski Geometries. London Mathematical Society Lecture Note Series, vol. 360 (Cambridge University Press, Cambridge, 2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.School of MathematicsUniversity of LeedsLeedsUK
  2. 2.School of Science and Technology, Division of MathematicsUniversity of CamerinoCamerinoItaly

Personalised recommendations