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Alfred Tarski's work in model theory

Abstract

We will consider Tarski's work in pure model theory and classical logic. His work in applied model theory—the model theory of various special theories—is discussed by Doner and van den Dries [1987], and McNulty [1986]. (However, the separation of “pure” and “applied” only becomes natural as the subjects mature; so we shall discuss applied model theory at least to some extent in Tarski's earlier work.)

Alfred Tarski (1901–1983) was awarded a Ph.D. in mathematics at Warsaw University in 1924. His teachers included the two leaders of the renowned Polish logic school, the logician-philosophers L. Leśniewski and J. Łukasiewicz. (Very soon Tarski was recognized as the third leader of the school.) Another teacher was the philosopher T. Kotarbiński, to whom Tarski dedicated his collected papers [56m]. Leśniewski was Tarski's thesis advisor; he transmitted to Tarski his interests in the metalanguage and in the theory of definition. Tarski's thesis ([23a], [24]) was about protothetic—the sentential calculus augmented by quantifiable variables ranging over truth functions. Its main result was that all the connectives are definable using only ↔ and ∀. By the same year, 1924, Tarski also had begun his prolific writings in set theory, and had discovered together with S. Banach, the leader of the Polish mathematicians, their famous “paradox” [24d] in measure theory. (For details see Lévy [1987].)

In 1927-29 Tarski held a seminar at Warsaw University on results he obtained in 1926-28. The seminar lay at the heart of what is now called model theory. The brief history of model theory up to that time had begun with the paper of L. Löwenheim [1915].

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(OTHER THAN WORKS OF TARSKI)

Addison J. W. [1957] Hierarchies and the axiom of constructibility, Summaries of talks presented at the summer institute for symbolic logic, Cornell University, 1957, 2nd ed., Communications Research Division, Institute for Defense Analyses, Princeton, New Jersey, 1960, pp. 355362.
Addison J. W. [1965] The undefinability of the definable, Notices of the American Mathematical Society, vol. 12, pp. 347348 (abstract 622-71).
Beth E. [1953] On Padoa's method in the theory of definition, Koninklijke Nederlandse Akademie tan Wetenschappen, Proceedings, Series A, vol. 56 = Indagationes Mathematicae, vol. 15, pp. 330339.
Birkhoff Garrett [1935] On the structure of abstract algebras, Proceedings of the Cambridge Philosophical Society, vol. 31, pp. 433454.
Blok W. J. and Pigozzi D. L. [1987] this Journal, vol. 52 (to appear).
Chang C. C. and Keisler H. J. [1973] Model theory, North-Holland, Amsterdam.
Doner J. and van den Dries L. [1987] this Journal, vol. 52 (to appear).
Etchemendy J. [1987] Tarski on truth and logical consequence, this Journal, vol. 52 (to appear).
Fraïssé R. [1954] Sur quelques classifications des systèmes de relations, Publications Scientifiques de l'Université d'Alger, Série A: Sciences Mathématiques, vol. 1, pp. 35182. (Also Thèse, Université de Paris, Imprimerie Durand, Chartres, 1955.)
Frayne T., Morel A. and Scott D. [1958] Reduced direct products, Notices of the American Mathematical Society, vol. 5, p. 674 (abstract 550-9).
Frayne T., Morel A. and Scott D. [1962] Reduced direct products, Fundamenta Mathematicae, vol. 51, pp. 195228.
Frayne T. and Scott D. [1958] Model-theoretic properties of reduced products, Notices of the American Mathematical Society, vol. 5, p. 675 (abstract 550-10).
Gödel K. [1930] Die Vollständigkeit der Axiome des logischen Funktionenkalküls, Monatshefte für Mathematik und Physik, vol. 37, pp. 349360.
Gödel K. [1931] Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. I, Monatshefte für Mathematik und Physik, vol. 38, pp. 173198.
Gödel K. [1934] Review of Skolem [1933], Zentralblatt für Mathematik und ihre Grenzgebiete, vol. 7, pp. 193194.
Hanf W. [1964] Incompactness in languages with infinitely long expressions, Fundamenta Mathematicae, vol. 53, pp. 309324.
Hanf W. and Scott D. [1961] Classifying inaccessible cardinals, Notices of the American Mathematical Society, vol. 8, p. 445 (abstract 61T-240.)
Henkin L. [1949] The completeness of the first-order functional calculus, this Journal, vol. 14, pp. 159166.
Jónsson B. [1986] The contributions of Alfred Tarski to general algebra, this Journal, vol. 51, pp. 883889.
Karp C. [1965] Languages with expressions of infinite length, North-Holland, Amsterdam.
Keisler H. J. [1961] Ultraproducts and elementary classes, Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series A, vol. 64 = Indagationes Mathematicae, vol. 23, pp. 477495.
Kleene S. C. [1938] On notation for ordinal numbers, this Journal, vol. 3, pp. 150155.
Kreisel G. [1957] Independent recursive axiomatization, this Journal, vol. 22, p. 109.
Kuratowski C. [1937] Les types d'ordre définissables et les ensembles boreliens, Fundamenta Mathematicae, vol. 28, pp. 97100.
Langford C. H. [1926] Some theorems on deducibility, Annals of Mathematics, ser. 2, vol. 28, pp. 1640.
Langford C. H. [1927] Theorems on deducibility, Annals of Mathematics, ser. 2, vol. 29, pp. 459471.
Lévy A. [1987] Alfred Tarski's contributions to set theory, this Journal, vol. 52 (to appear).
Łoś J. [1955] Quelques remarques, théorèmes et problèmes sur les classes définissables d'algèbres, Mathematical interpretation of formal systems, North-Holland, Amsterdam, pp. 98113.
Łoś J. [1955a] On the extending of models. I, Fundamenta Mathematicae, vol. 42, pp. 3854.
Löwenheim L. [1915] Über Möglichkeiten im Relativkalkül, Mathematische Annalen, vol. 76, pp. 447470.
Lyndon R. [1950] The representation of relational algebras, Annals of Mathematics, ser. 2, vol. 51, pp. 707729.
McNulty G. F. [1986] Alfred Tarski and undecidable theories, this Journal, vol. 51, pp. 890898.
Makkai M. [1964] On PCΔ-classes in the theory of models, A Magyar Tudományos Akadémia Matematikai Kutató Intézetének Közleményei, ser. A, vol. 9, pp. 159–194, 601602.
Mal'cev A. [1936] Untersuchungen aus dem Gebiete der mathematischen Logik, Matematičeskiǐ Sbornik, vol. 1 (43), pp. 323336.
Mal'cev A. [1941] On a general method for obtaining local theorems in group theory, Ivanovskiǐ Gosudarstvennyǐ Pedagogičeskiǐ Institut, Učenye Zapiski, Fiziko-Matematičeskie Nauki, vol. 1, pp. 39 (Russian); English transl., Chapter II in A. I.|Mal'cev, The metamathematics of algebraic systems. Collected papers: 1936–1967, North-Holland, Amsterdam, 1971, pp. 15–21.
Monk J. D. [1986] The contributions of Alfred Tarski to algebraic logic, this Journal, vol. 51, pp. 899906.
Montague R. and Kalish D. [1956] A simplification of Tarski's formulation of the predicate calculus, Bulletin of the American Mathematical Society, vol. 62, p. 261.
Mostowski A. [1937] Abzahlbäre Boolesche Körper und ihre Anwendung auf die allgemeine Metamathematik, Fundamenta Mathematicae, vol. 29, pp. 3453.
Presburger M. [1930] Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt, Comptes-rendus du I congrès des mathématiciens des pays slaves, Warszawa, 1929, Ksiażnica Atlas T.N.S.W. Warsaw, 1930, pp. 92-101, 395.
Reznikoff I. [1965] Tout ensemble de formules de la logique classique est équivalent à un ensemble indépendant, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, vol. 260, pp. 23852388.
Robinson A. [1951] On the metamathematics of algebra, North-Holland, Amsterdam.
Rosser J. B. [1937] Godei theorems for non-constructive logics, this Journal, vol. 2, pp. 129137.
Sacerdote G. S. [1973] Elementary properties of free groups, Transactions of the American Mathematical Society, vol. 178, pp. 127138.
Scott D. [1961] Measurable cardinals and constructible sets, Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 9, pp. 521524.
Scott D. [1965] Logic with denumerably long formulas and finite strings of quantifiers, Proceedings of the 1963 Berkeley symposium on the theory of models (Addison J. et al., editors), North-Holland, Amsterdam, pp. 327341.
Shelah S. [1971] Every two elementarily equivalent models have isomorphic ultrapowers, Israel Journal of Mathematics, vol. 10, pp. 224233.
Skolem T. [1919] Untersuchungen über die Axiome des Klassenkalkuls und über Produktations- und Sum-mationsprobleme, welche gewisse Klassen von Aussagen betreffen, Skrifter utgit av Videnskapsselskapet i Kristiania. I, Matematisk-Naturvidenskabelig Klasse, no. 3.
Skolem T. [1920] Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen, Skrifter utgit av Videnskapsselskapet i Kristiania. I, Matematisk-Naturvidenskabelig Klasse, no. 4.
Skolem T. [1922] Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre, Wissenschaftliche Vorträge gehalten auf dem fünften Kongress der Skandinavischen Mathematiker in Helsingfors vom 4. bis 7. Juli 1922, Akademische Buchhandlung, Helsinki, 1923, pp. 217232.
Skolem T. [1933] Über die Unmöglichkeit einer vollständigen Charakterisierung der Zahlenreihe mittels eines endlichen Axiomensystems, Norsk Matematisk Forenings Skrifter, ser. 2, no. 10, pp. 7382.
Skolem T. [1934] Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen, Fundamenta Mathematicae, vol. 23, pp. 150161.
Stone M. H. [1936] The theory of representations for Boolean algebras, Transactions of the American Mathematical Society, vol. 40, pp. 37111.
Stone M. H. [1937] Applications of the theory of Boolean rings to general topology, Transactions of the American Mathematical Society, vol. 41, pp. 375481.
Suppes P. F. [1987] this Journal, vol. 52 (to appear).
Vaught R. [1961] The elementary character of two notions from general algebra, Essays on the foundations of mathematics, dedicated to A. A. Fraenkel on his seventieth anniversary (Bar-Hillel Y. et al., editors), Magnus Press, Jerusalem, pp. 226233.
Vaught R. [1965] A Löwenheim-Skolem theorem for cardinals far apart, Proceedings of the 1963 Berkeley symposium on the theory of models (Addison J. et al., editors), North-Holland, Amsterdam, pp. 390401.
Vaught R. [1974] Model theory before 1945, Proceedings of the Tarski symposium, Proceedings of Symposia in Pure Mathematics, vol. 25, American Mathematical Society, Providence, Rhode Island, pp. 153172.
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The Journal of Symbolic Logic
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  • EISSN: 1943-5886
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