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Alfred Tarski and decidable theories

Published online by Cambridge University Press:  12 March 2014

John Doner  and
Wilfrid Hodges
John Doner
Affiliation:
Department of Mathematics, University of California at santa Barbara, Santa Barbara, California 93106
Wilfrid Hodges
Affiliation:
School of Mathematical Sciences, Queen Mary College, London El 4NS, England
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Extract

Any list of Alfred Tarski's achievements would mention his decision procedure for real-closed fields. He proved a number of other less publicized decidability results too. We shall survey these results. After surveying them we shall ask what Tarski had in mind when he proved them. Today our emphases and concepts are sometimes different from those of Tarski in the early 1930s. Some of these changes are the direct result of Tarski's own fundamental work in model theory during the intervening years.

Tarski's work on decidable theories is important not just for the individual decidability theorems themselves. His method for all these decidability results was elimination of quantifiers, and he systematically used this method to prove a range of related theorems about completeness and definability. He also led several of his students to do important work using this same method. Tarski's use of quantifier elimination has had a deep and cumulative influence on model theory and the logical treatment of algebraic theories.

We thank Solomon Feferman, Steven Givant, Haragauri Gupta, Yuri Gurevich. Angus Macintyre, Gregory Moore, Robert Vaught and the referee for helpful discussions and comments. Also we thank Madame Maria Mostowska and Roman Murawski for sending us material from Polish libraries.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 1 , March 1988 , pp. 20 - 35
Copyright
Copyright © Association for Symbolic Logic 1988

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References

REFERENCES

Blok, W. J. and Pigozzi, Don [1988] Alfred Tarski's work on general metamathematics, this Journal, vol. 53, pp. 36–50.Google Scholar
Chang, C. C. and Keisler, H. J. [1973] Model theory, North-Holland, Amsterdam.Google Scholar
Church, A. [1936] An unsolvable problem of elementary number theory, American Journal of Mathematics, vol. 58, pp. 345–363.CrossRefGoogle Scholar
Doner, J. E. [1970] Tree acceptors and some of their applications, Journal of Computer and System Sciences, vol. 4, pp. 406–451.CrossRefGoogle Scholar
van den Dries, Lou [1988] Alfred Tarski's elimination theory for real closed fields, this Journal, vol. 53, pp. 7–19.Google Scholar
Dyson, Verena Huber [1964] On the decision problem for theories of finite models, Israel Journal of Mathematics, vol. 2, pp. 55–70.CrossRefGoogle Scholar
Ershov, Yu. L. [1964] Decidability of the elementary theory of distributive lattices with relative complements and the theory of filters, Algebra i Logika, vol. 3, no. 3, pp. 17–38. (Russian)Google Scholar
Ershov, Yu. L. [1980] Problems of decidability and constructive models, “Nauka”, Moscow. (Russian)Google Scholar
Feferman, S. and Vaught, R. L. [1959] The first order properties of algebraic systems, Fundamenta Mathematicae, vol. 47, pp. 57–103.CrossRefGoogle Scholar
Ferrante, Jeanne and Rackoff, Charles W. [1979] The computational complexity of logical theories, Lecture Notes in Mathematics, vol. 718, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Fischer, Michael J. and Rabin, Michael O. [1974] Super-exponential complexity of Presburger arithmetic, Complexity of computation, (Karp, Richard M., editor), SIAM-AMS Proceedings, vol. 7, American Mathematical Society, Providence, Rhode Island, pp. 27–41.Google Scholar
Fraenkel, A. [1928] Einleitung in die Mengenlehre, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Gödel, K. [1986] Collected works. Vol. 1 (Feferman, Solomon et al., editors), Oxford University Press, Oxford.Google Scholar
Gurevich, Y. [1985] Monadic second-order theories, Model-theoretic logics (Barwise, J. and Feferman, S., editors), Springer-Verlag, New York, pp. 479–506.Google Scholar
Hermann, Grete [1926] Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Mathematische Annalen, vol. 95, pp. 736–788.CrossRefGoogle Scholar
Hilbert, D. and Ackermann, W. [1928] Grundzüge der theoretischen Logik, Springer-Verlag, Berlin (2nd éd., 1938).Google Scholar
Kleene, Stephen Cole [1943] Recursive sets and quantifiers, Transactions of the American Mathematical Society, vol. 53, pp. 41–73.CrossRefGoogle Scholar
Langford, C. H. [1927] Some theorems on deducibility, Annals of Mathematics, ser. 2, vol. 28, pp. 16–40.CrossRefGoogle Scholar
Langford, C. H. [1927a] Theorems on deducibility (second paper), Annals of Mathematics, ser. 2, vol. 28, pp. 459–471.CrossRefGoogle Scholar
Lewis, C. I. and Langford, C. H. [1932] Symbolic logic, Century, New York; reprint, Dover, New York, 1959.Google Scholar
Löwenheim, Leopold [1915] Über Möglichkeiten im Relativkalkül, Mathematische Annalen, vol. 76, pp. 447–470; English translation in J. van Heijenoort (editor), From Frege to Gödel , Harvard University Press, Cambridge, Massachusetts, 1967, pp. 228–251.CrossRefGoogle Scholar
Mal′cev, A. I. [1962] Axiomatizable classes of locally free algebras of certain types, Sibirskiĭ Matematicheskiĭ Zhurnal, vol. 3, pp. 729–743; English translation, Chapter XXIII in A. I. Mal′cev, The metamathematics of algebraic systems. Collected papers, 1936–1967 , North-Holland, Amsterdam, 1971.Google Scholar
Mcnulty, George F. [1986] Alfred Tarski and undecidable theories, this Journal, vol. 51, pp. 890–898.Google Scholar
Mekler, Alan H. [1984] Stationary logic of ordinals, Annals of Pure and Applied Logic, vol. 26, pp. 47–68.CrossRefGoogle Scholar
Monk, J. Donald [1986] The contributions of Alfred Tarski to algebraic logic, this Journal, vol. 51, pp. 899–906.Google Scholar
Morley, Michael [1965] Categoricity in power, Transactions of the American Mathematical Society, vol. 114, pp. 514–538.CrossRefGoogle Scholar
Pillay, Anand And Steinhorn, Charles [1986] Definable sets in ordered structures. I, Transactions of the American Mathematical Society, vol. 295, pp. 565–592.CrossRefGoogle Scholar
Presburger, M. [1930] Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt, Sprawozdanie z I Kongresu Matematików Krajów Shwiańskich ( = Comptes-rendus du I Congrès des Mathématiciens des Pays Slates), Warsaw, pp. 92–101.Google Scholar
Robinson, Abraham [1958] Relative model-completeness and the elimination of quantifiers, Dialectica, vol. 12, pp. 146–159.CrossRefGoogle Scholar
Shelah, Saharon [1978] Classification theory and the number of non-isomorphic models, North-Holland, Amsterdam.Google Scholar
Shoenfield, J. R. [1971] A theorem on quantifier elimination, Symposia Mathematica, vol. V (INDAM, Rome, 1969/1970), Academic Press, London, pp. 173–176.Google Scholar
Skolem, Th. [1919] Untersuchungen über die Axiome des Klassenkalküls und über Produktations- und Summationsprobleme, welche gewisse Klassen von Aussagen betreffen, Skrifter Videnskapsakademiet i Kristiania, vol. 3, pp. 37–71.Google Scholar
Skolem, Th. [1928] Über die mathematische Logik, Norsk Mathematisk Tidsskrift, vol. 10, pp. 125–142; English translation in J. van Heijenoort (editor), From Frege to Gödel , Harvard University Press, Cambridge, Massachusetts, 1967, pp. 508–524.Google Scholar
Skolem, Th. [1930] Über einige Satzfunktionen in der Arithmetik, Skrifter Videnskapsakademiet i Oslo I, no. 7.Google Scholar
Szczerba, L. W. [1986] Tarski and geometry, this Journal, vol. 51, pp. 907–912.Google Scholar
Szmielew, Wanda [1949] Arithmetical classes and types of Abelian groups, Bulletin of the American Mathematical Society, vol. 55, pp. 65, 1192.Google Scholar
Szmielew, Wanda [1955] Elementary properties of Abelian groups, Fundamenta Mathematicae, vol. 41, pp. 203–271.CrossRefGoogle Scholar
Vaught, Robert L. [1986] Alfred Tarski's work in model theory, this Journal, vol. 51, pp. 869–882.Google Scholar