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).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Cohn, P.M.: Basic Algebra. Springer, London (2003)
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)
Goldblatt, R.: Lectures on the Hyperreals – An Introduction to Nonstandard Analysis. Graduate Texts in Mathematics, vol. 188. Springer, New York (1998)
Kreisel, G., Krivine, J.L.: Elements of Mathematical Logic. Studies in Logic. North-Holland, Amsterdam (1967)
Lang, S.: Algebra. Graduate Texts in Mathematics, vol. 211. Springer, Princeton (2002)
Marker, D.: Model Theory – An Introduction. Graduate Texts in Mathematics, vol. 217 Springer, Berlin (2002)
Marker, D., Messmer, M., Pillay, A.: Model Theory of Fields. Lecture Notes in Logic, vol. 5, 2nd edn. Association for Symbolic Logic (2002)
Prawitz, D.: Natural Deduction – A Proof-Theoretical Study. Dover, Mineola (2006); Reprint of 1965 Ph.D. thesis
Robinson, A.: Non-standard Analysis. North-Holland, Amsterdam (1966)
Tarski, A.: Der Wahrheitsbegriff in den formalisierten Sprachen. Studia Philosophica 1, 261–405 (1935)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Moerdijk, I., van Oosten, J. (2018). Models. In: Sets, Models and Proofs. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-92414-4_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-92414-4_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-92413-7
Online ISBN: 978-3-319-92414-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)