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Alfred Tarski and decidable theories

  • John Doner (a1) and Wilfrid Hodges (a2)


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.



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Blok, W. J. and Pigozzi, Don [1988] Alfred Tarski's work on general metamathematics, this Journal, vol. 53, pp. 3650.
Chang, C. C. and Keisler, H. J. [1973] Model theory, North-Holland, Amsterdam.
Church, A. [1936] An unsolvable problem of elementary number theory, American Journal of Mathematics, vol. 58, pp. 345363.
Doner, J. E. [1970] Tree acceptors and some of their applications, Journal of Computer and System Sciences, vol. 4, pp. 406451.
van den Dries, Lou [1988] Alfred Tarski's elimination theory for real closed fields, this Journal, vol. 53, pp. 719.
Dyson, Verena Huber [1964] On the decision problem for theories of finite models, Israel Journal of Mathematics, vol. 2, pp. 5570.
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. 1738. (Russian)
Ershov, Yu. L. [1980] Problems of decidability and constructive models, “Nauka”, Moscow. (Russian)
Feferman, S. and Vaught, R. L. [1959] The first order properties of algebraic systems, Fundamenta Mathematicae, vol. 47, pp. 57103.
Ferrante, Jeanne and Rackoff, Charles W. [1979] The computational complexity of logical theories, Lecture Notes in Mathematics, vol. 718, Springer-Verlag, Berlin.
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. 2741.
Fraenkel, A. [1928] Einleitung in die Mengenlehre, Springer-Verlag, Berlin.
Gödel, K. [1986] Collected works. Vol. 1 (Feferman, Solomonet al., editors), Oxford University Press, Oxford.
Gurevich, Y. [1985] Monadic second-order theories, Model-theoretic logics (Barwise, J. and Feferman, S., editors), Springer-Verlag, New York, pp. 479506.
Hermann, Grete [1926] Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Mathematische Annalen, vol. 95, pp. 736788.
Hilbert, D. and Ackermann, W. [1928] Grundzüge der theoretischen Logik, Springer-Verlag, Berlin (2nd éd., 1938).
Kleene, Stephen Cole [1943] Recursive sets and quantifiers, Transactions of the American Mathematical Society, vol. 53, pp. 4173.
Langford, C. H. [1927] Some theorems on deducibility, Annals of Mathematics, ser. 2, vol. 28, pp. 1640.
Langford, C. H. [1927a] Theorems on deducibility (second paper), Annals of Mathematics, ser. 2, vol. 28, pp. 459471.
Lewis, C. I. and Langford, C. H. [1932] Symbolic logic, Century, New York; reprint, Dover, New York, 1959.
Löwenheim, Leopold [1915] Über Möglichkeiten im Relativkalkül, Mathematische Annalen, vol. 76, pp. 447470; English translation in J. van Heijenoort (editor), From Frege to Gödel, Harvard University Press, Cambridge, Massachusetts, 1967, pp. 228–251.
Mal′cev, A. I. [1962] Axiomatizable classes of locally free algebras of certain types, Sibirskiĭ Matematicheskiĭ Zhurnal, vol. 3, pp. 729743; English translation, Chapter XXIII in A. I. Mal′cev, The metamathematics of algebraic systems. Collected papers, 1936–1967, North-Holland, Amsterdam, 1971.
Mcnulty, George F. [1986] Alfred Tarski and undecidable theories, this Journal, vol. 51, pp. 890898.
Mekler, Alan H. [1984] Stationary logic of ordinals, Annals of Pure and Applied Logic, vol. 26, pp. 4768.
Monk, J. Donald [1986] The contributions of Alfred Tarski to algebraic logic, this Journal, vol. 51, pp. 899906.
Morley, Michael [1965] Categoricity in power, Transactions of the American Mathematical Society, vol. 114, pp. 514538.
Pillay, Anand And Steinhorn, Charles [1986] Definable sets in ordered structures. I, Transactions of the American Mathematical Society, vol. 295, pp. 565592.
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. 92101.
Robinson, Abraham [1958] Relative model-completeness and the elimination of quantifiers, Dialectica, vol. 12, pp. 146159.
Shelah, Saharon [1978] Classification theory and the number of non-isomorphic models, North-Holland, Amsterdam.
Shoenfield, J. R. [1971] A theorem on quantifier elimination, Symposia Mathematica, vol. V (INDAM, Rome, 1969/1970), Academic Press, London, pp. 173176.
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. 3771.
Skolem, Th. [1928] Über die mathematische Logik, Norsk Mathematisk Tidsskrift, vol. 10, pp. 125142; English translation in J. van Heijenoort (editor), From Frege to Gödel, Harvard University Press, Cambridge, Massachusetts, 1967, pp. 508–524.
Skolem, Th. [1930] Über einige Satzfunktionen in der Arithmetik, Skrifter Videnskapsakademiet i Oslo I, no. 7.
Szczerba, L. W. [1986] Tarski and geometry, this Journal, vol. 51, pp. 907912.
Szmielew, Wanda [1949] Arithmetical classes and types of Abelian groups, Bulletin of the American Mathematical Society, vol. 55, pp. 65, 1192.
Szmielew, Wanda [1955] Elementary properties of Abelian groups, Fundamenta Mathematicae, vol. 41, pp. 203271.
Vaught, Robert L. [1986] Alfred Tarski's work in model theory, this Journal, vol. 51, pp. 869882.


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