Abstract
The axiomatization presented in the previous section is an attempt to axiomatize the set of theorems true in the intended structure. How well does it do this? The best for which one can hope in an axiomatization is that it can serve as the basis for an effective decision procedure, i.e. a procedure which, given any sentence S of the language, decides in finitely many steps whether or not S is true in the intended structure. The most informative decision procedures are those which proceed by quantifier elimination. This method is also the oldest. It was applied in the twenties by Langford to the theory of dense orderings, by Presburger to the (additive) theory of the integers, and finally by Tarski to the theory that we are considering. The paper by Yu. Ershov, mentioned in the references, surveys numerous subsequent applications of the method.
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Further Reading
Tarski, A.: A Decision Method of Elementary Algebra and Geometry, Berkeley, University of California Press, 1951
Seidenberg, A.: A New Decision Method for Elementary Algebra, Annals of Mathematics, vol. 60, PP. 365–374, (1954)
Ershov, Yu. et. al.: Elementary Theories, Russian Mathematical Surveys, vol. 20 (4), pp. 35–105, in particular pp. 94–99, (1965)
Fischer, M.J. & Rabin, M.O.: Super-Exponential Complexity of Presburger Arithmetic, SIAM-AMS Proceedings, vol. 7 pp. 27–41, (1974)
Collins, G.E.: Quantifier Elimination for Real Closed Fields by Cylindrical Algebra Decomposition, Springer Lecture Notes in Computer Science 33, pp. 134–183, (1975)
Hilbert, D.: Mathematische Probleme, 17. Problem: Darstellung definiter Formen durch Quadrate, Ostwalds Klassiker der exakten Wissenschaften, vol. 252, pp. 63–64. Leipzip, Akad. Verlagsgesellschaft, 1933
Artin, E. & Schreier, O.: Algebraische Konstruktion reeller Körper, Abhandlungen aus dem mathematischen Seminar der Universität Hamburg, vol. 5, pp. 85–99, (1927)
Artin, E.: Ueber die Zerlegung definiter Funktionen in Quadrate, Abhandlungen aus dem mathematischen Seminar der Universität Hamburg, vol. 5 pp. 100–115, (1927)
Friedmann, A.: Generalized Functions and Partial Differential Equations, in particular pp. 218–225. Englewood Cliffs, Prentice-Hall, 1963
Heintz, J., Roy, M.-F., Solernó, P.: On the complexity of semialgebraic sets, Proc. IFIP (San Francisco 1989), North Holland, pp. 293–298, (1989)
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© 1993 Springer-Verlag Berlin Heidelberg
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Engeler, E. (1993). Elementary Theory of Real Numbers. In: Foundations of Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-78052-3_3
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