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Models

  • Ieke Moerdijk
  • Jaap van Oosten
Chapter
Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Abstract

Formal languages, structures and theories are introduced. Definability issues are presented. The Compactness Theorem is given, with many applications. The compactness is proved using ultrafilters. Elements of model theory are presented: quantifier elimination (as an example: QE for algebraically closed fields, with applications to commutative algebra), elementary substructures, Löwenheim–Skolem theorems and ω-categoricity (back-and-forth method).

References

  1. 11.
    Cohn, P.M.: Basic Algebra. Springer, London (2003)CrossRefGoogle Scholar
  2. 15.
    de Bruijn, N.G., Erdős, P.: A colour problem for infinite graphs and a problem in the theory of relations. Indagationes Mathematicae 13, 369–373 (1951)Google Scholar
  3. 25.
    Goldblatt, R.: Lectures on the Hyperreals – An Introduction to Nonstandard Analysis. Graduate Texts in Mathematics, vol. 188. Springer, New York (1998)Google Scholar
  4. 45.
    Kreisel, G., Krivine, J.L.: Elements of Mathematical Logic. Studies in Logic. North-Holland, Amsterdam (1967)Google Scholar
  5. 47.
    Lang, S.: Algebra. Graduate Texts in Mathematics, vol. 211. Springer, Princeton (2002)CrossRefGoogle Scholar
  6. 49.
    Marker, D.: Model Theory – An Introduction. Graduate Texts in Mathematics, vol. 217 Springer, Berlin (2002)Google Scholar
  7. 51.
    Marker, D., Messmer, M., Pillay, A.: Model Theory of Fields. Lecture Notes in Logic, vol. 5, 2nd edn. Association for Symbolic Logic (2002)Google Scholar
  8. 62.
    Prawitz, D.: Natural Deduction – A Proof-Theoretical Study. Dover, Mineola (2006); Reprint of 1965 Ph.D. thesisGoogle Scholar
  9. 64.
    Robinson, A.: Non-standard Analysis. North-Holland, Amsterdam (1966)zbMATHGoogle Scholar
  10. 75.
    Tarski, A.: Der Wahrheitsbegriff in den formalisierten Sprachen. Studia Philosophica 1, 261–405 (1935)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Ieke Moerdijk
    • 1
  • Jaap van Oosten
    • 1
  1. 1.Department of MathematicsUtrecht UniversityUtrechtThe Netherlands

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