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Alfred Tarski's work in model theory


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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K. Gödel [1930] Die Vollständigkeit der Axiome des logischen Funktionenkalküls, Monatshefte für Mathematik und Physik, vol. 37, pp. 349360.

C. H. Langford [1926] Some theorems on deducibility, Annals of Mathematics, ser. 2, vol. 28, pp. 1640.

J. Łoś [1955] Quelques remarques, théorèmes et problèmes sur les classes définissables d'algèbres, Mathematical interpretation of formal systems, North-Holland, Amsterdam, pp. 98113.

L. Löwenheim [1915] Über Möglichkeiten im Relativkalkül, Mathematische Annalen, vol. 76, pp. 447470.

R. Lyndon [1950] The representation of relational algebras, Annals of Mathematics, ser. 2, vol. 51, pp. 707729.

G. S. Sacerdote [1973] Elementary properties of free groups, Transactions of the American Mathematical Society, vol. 178, pp. 127138.

S. Shelah [1971] Every two elementarily equivalent models have isomorphic ultrapowers, Israel Journal of Mathematics, vol. 10, pp. 224233.

M. H. Stone [1936] The theory of representations for Boolean algebras, Transactions of the American Mathematical Society, vol. 40, pp. 37111.

M. H. Stone [1937] Applications of the theory of Boolean rings to general topology, Transactions of the American Mathematical Society, vol. 41, pp. 375481.

R. Vaught [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
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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