Elements of Model Theory

Part of the Universitext book series (UTX)


Model theory is a main branch of applied mathematical logic. Here the techniques developed in mathematical logic are combined with construction methods of other areas (such as algebra and analysis) to their mutual benefit. The following demonstrations can provide only a first glimpse in this respect, a deeper understanding being gained, for instance, from [CK] or [Ho]. For further-ranging topics, such as saturated models, stability theory, and the model theory of languages other than first-order, we refer to the special literature, [Bue], [Mar], [Pz], [Rot], [Sa], [She].

The theorems of Lwenheim and Skolem were first formulated in the generality given in 5.1 by Tarski. These and the compactness theorem form the basis of model theory, a now wide-ranging discipline that arose around 1950. Key concepts of model theory are elementary equivalence and elementary extension. These not only are interesting in themselves but also have multiple applications to model constructions in set theory, nonstandard analysis, algebra, geometry and elsewhere.


Winning Strategy Edge Element Elementary Extension Quotient Field Quantifier Elimination 


  1. CK.
    C. C. Chang, H. J. Keisler, Model Theory, Amsterdam 1973, 3⟨{ rd}⟩ ed. North-Holland 1990.Google Scholar
  2. Ho.
    W. Hodges, Model Theory, Cambridge Univ. Press 1993.Google Scholar
  3. Bue.
    S. Buechler, Essential Stability Theory, Springer 1996.Google Scholar
  4. Mar.
    D. Marker, Model Theory, An Introduction, Springer 2002.Google Scholar
  5. Pz.
    B. Poizat, A Course in Model Theory, Springer 2000.Google Scholar
  6. Rot.
    P. Rothmaler, Introduction to Model Theory, Gordon & Breach 2000.Google Scholar
  7. Sa.
    G. Sacks, Saturated Model Theory, W. A. Benjamin 1972.Google Scholar
  8. She.
    S. Shelah, Classification Theory and the Number of Nonisomorphic Models, Amsterdam 1978, 2⟨{ nd}⟩ ed. North-Holland 1990.Google Scholar
  9. De.
    O. Deiser, Axiomatische Mengenlehre, Springer, to appear 2010.Google Scholar
  10. Ku.
    K. Kunen, Set Theory, An Introduction to Independence Proofs, North-Holland 1980.Google Scholar
  11. HR.
    H. Herre, W. Rautenberg, Das Basistheorem und einige Anwendungen in der Modelltheorie, Wiss. Z. Humboldt-Univ., Math. Nat. R. 19 (1970), 579–583.Google Scholar
  12. MV.
    R. McKenzie, M. Valeriote, The Structure of Decidable Locally Finite Varieties, Progress in Mathematics 79, Birkhuser 1989.Google Scholar
  13. Wae.
    B. L. van der Waerden, Algebra I, Berlin 1930, 4⟨{ th}⟩ ed. Springer 1955.Google Scholar
  14. Zi.
    M. Ziegler, Model theory of modules, Ann. Pure Appl. Logic 26 (1984), 149–213.Google Scholar
  15. Pr.
    M. Presburger, Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt, Congrès des Mathématiciens des Pays Slaves 1 (1930), 92–101.Google Scholar
  16. Wi.
    A. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, Journal Amer. Math. Soc. 9 (1996), 1051–1094.Google Scholar
  17. Dav.
    M. Davis (editor), The Undecidable, Raven Press 1965.Google Scholar
  18. Ro1.
    A. Robinson, Introduction to Model Theory and to the Metamathematics of Algebra, Amsterdam 1963, 2⟨{ nd}⟩ ed. North-Holland 1974.Google Scholar
  19. Wae.
    B. L. van der Waerden, Algebra I, Berlin 1930, 4⟨{ th}⟩ ed. Springer 1955.Google Scholar
  20. Ta3.
    _________ , A Decision Method for Elementary Algebra and Geometry, Santa Monica 1948, Berkeley 1951, Paris 1967.Google Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Fachbereich Mathematik und InformatikBerlinGermany

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