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Unifying threads in Alfred Tarski’s work. (English) Zbl 0922.03004

The work of Alfred Tarski covers a great range of subjects: set theory, measure theory, topology, geometry, classical and universal algebra, algebraic logic, various branches of formal logic and metamathematics. The aim of the paper under review is to find the underlying unity in Tarski’s work and to trace steps that may have led to some of his discoveries.

MSC:

03-03 History of mathematical logic and foundations
01A60 History of mathematics in the 20th century

Biographic References:

Tarski, Alfred
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[1] Stefan Banach and Alfred Tarski, ”Sur la décomposition des ensembles de points en parties respectivement congruentes,”Fundamenta Mathematicae 6, 1924, 244–277. · JFM 50.0370.02
[2] Garrett Birkhoff, ”On the combination of subalgebras,”Proceedings of the Cambridge Philosophical Society 29, 1933, 441–464. · JFM 59.0154.02 · doi:10.1017/S0305004100011464
[3] Louise Chin and Alfred Tarski, ”Distributive and modular laws in the arithmetic of relation algebras”,University of California Publications in Mathematics, new series, vol. 1, 1951, 341–384. · Zbl 0045.31701
[4] Krzysztof Ciesielski and Zdzisw Pogoda, ”The beginning of Polish topology,”Mathematical Intelligencer 18, 1996, no. 3, 32- 39. · Zbl 0881.54002 · doi:10.1007/BF03024308
[5] August De Morgan, ”On the syllogism: IV, and on the logic of relations,”Transactions of the Cambridge Philosophical Society 10, 1864, 331–358.
[6] Arnaud Denjoy,L’énumération transfinie, Paris: Gauthier-Villars, 1954. · Zbl 0056.04702
[7] John Donor, Andrzej Mostowski, and Alfred Tarski, ”The elemen- tary theory of well-ordering-A metamathematical study,” in: Angus Maclntyre, Leszek Pacholski, and Jeff Paris (Eds.),Logic Colloquium 77, Amsterdam: North-Holland Publishing Company, 1978, pp. 1–54. · Zbl 0461.03003
[8] Editorial remark,Fundamenta Mathematicae 23, 1934, 161.
[9] Howard Eves and Carroll Newsom,An introduction to the foundations and fundamental concepts of mathematics, New York: Holt, Rinehart, and Winston, 1958. · Zbl 0084.00308
[10] Robert French, ”The Banaoh-Tarski paradox,”Mathematical Intelligencer 10, 1988, no. 4, 21–28. · Zbl 0657.51010 · doi:10.1007/BF03023740
[11] Steven Givant, ”A portrait of Alfred Tarski,”Mathematical Intelligencer 13, 1991, no. 3, 16–32. · Zbl 0733.01022 · doi:10.1007/BF03023831
[12] Felix Hausdorff,Grundzüge der Mengenlehre, Leipzig: Veit, 1914. (Reprinted: Bronx NY: Chelsea Publishing Company, 1949.)
[13] David Hubert,Grundlagen der Geometrie, Leipzig: Teubner, 1899.
[14] Leon Henkin, J. Donald Monk, and Alfred Tarski,Cylindric algebras. Part I, Amsterdam: North-Holland Publishing Company, 1971. · Zbl 0576.03043
[15] Leon Henkin, J. Donald Monk, Alfred Tarski, Hajnal Andréka, and Istv án Németi,Cylindric set algebras, Berlin: Springer Verlag, 1981.
[16] Leon Henkin, J. Donald Monk, and Alfred Tarski,Cylindric algebras, Park II, Amsterdam: North-Holland Publishing Company, 1985. · Zbl 0576.03043
[17] Bjarni Jönsson, ”A survey of Boolean algebras with operators”, in: Ivo Rosenberg and Gert Sabidussi (Eds.),Algebras and orders, NATO ASI Series C: Mathematical and Physical Sciences, vol. 389, Dortrecht: Kluwer Academic Publishers, 1993, pp. 239–286 · Zbl 0811.06012
[18] Bjarni Jónsson and Alfred Tarski, ”Boolean algebras with operators. Part I,”American Journal of Mathematics 73, 1951, 891–939. · Zbl 0045.31505 · doi:10.2307/2372123
[19] Bjarni Jönsson and Alfred Tarski, ”Boolean algebras with operators. Part II,”American Journal of Mathematics 74,1952, 127–162 (The abstract appeared as ”Representation problems for relation algebras”,Bulletin of the American Mathematical Society 54, 1948, 80.) · Zbl 0049.15801 · doi:10.2307/2372074
[20] Tadeusz Kotarbihski,Elementy teorji posnania, logiki formalnej i metodologji nauk, Lwöw: Wydawnictwo Zakfadu Narodowego im Ossoliriskich, 1929.
[21] Adolf Lindenbaum and Alfred Tarski, ”Communication sur les recherches de la théorie des ensembles,”Sprawozdania z Posiedzeh Towarzystwa Naukowego Warszawskiego, Wydziaf III Nauk Matematycznych i Przyrodniczych (=Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III) 19, 1926, 299–330.
[22] Adolf Lindenbaum and Alfred Tarski, ”Über Beschränktheit der Ausdrucksmittel deduktiver Theorien,”Ergebnisse eines Mathematischen Kolloquiums 7,1936, 15–22. (An English translation appeared as Article XIII in [66].) · Zbl 0014.38602
[23] Leopold Löwenheim, ”Über die Möglichkeiten im Relativkalkül,”Mathematische Annalen 76, 1915, 447–470. · JFM 45.0108.01 · doi:10.1007/BF01458217
[24] Leopold Löwenheim, ”Einkleidung der Mathematik in Schröderschen Relativkalkul,”Journal of Symbolic Logic 5, 1940, 1–15. · Zbl 0024.00202 · doi:10.2307/2269177
[25] Jan -Łukasiewicz and Alfred Tarski, ”Untersuchungen über den Aussagenkalkül”,Sprawozdania z Posiedzeh Towarzystwa Naukowego Warszawskiego, Wydziaf III Nauk Matematycznofizycznych (=Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III) 23, 1930, 30–50. (An English translation appeared as Article IV in [66].) · JFM 57.1319.01
[26] Roger Lyndon, ”The representation of relational algebras,”Annals of Mathematics 51, 1950, 707–729. · Zbl 0037.29302 · doi:10.2307/1969375
[27] Roger Lyndon, ”The representation of relation algebras, II,”Annals of Mathematics 63, 1956, 294–307. · Zbl 0070.24601 · doi:10.2307/1969611
[28] Gregory Moore,Zermelo’s axiom of choice: its origins, development, and influence, Berlin: Springer Verlag, 1982. · Zbl 0497.01005
[29] Charles Saunders Peirce, ”Note B: the logic of relatives,” in: Charles Saunders Peirce (Ed.),Studies in logic by the members of the Johns Hopkins University, Boston: Little, Brown, and Company, 1883.
[30] Mario Pieri, ”La geometria elementare istituita sulle nozioni ’punto’ é ’sfera’,”Memorie di Matematica e di Fisica délia Società Italiana delle Scienze 15, 1908, 345–450. · JFM 39.0545.08
[31] Mojzesz Presburger, ”Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt,”Sprawozdania z Pierwszego Kongresu Matematyków Krajów Sfowianskich (=Comptes- Rendus du Premier Congrès des Mathématiciens des Pays Slaves), Warsaw, 1930, pp. 92–101. · JFM 56.0825.04
[32] Andrzej Schinzel,Waolaw Sierpihski, Warsaw: Iskry, 1976.
[33] Ernst Schröder,Vorlesungen über die Algebra der Logik, vol. 3,Algebra und Logik der Relative, Band I, Leipzig: Teubner, 1895. (Reprinted: Bronx NY: Chelsea Publishing Company, 1966.) · Zbl 0188.30901
[34] Wacw Sierpiński,Zarys teoryi mnogości, Warsaw: Biblioteka Matematyczno-Fizyczna, 1912.
[35] Marshall Stone, ”On the structure of Boolean algebras,”Bulletin of the American Mathematical Society 39, 1933, 200.
[36] Marshall Stone, ”Boolean algebras and their application to topology,”Proceedings of the National Academy of Sciences 20, 1934, 197–202. · Zbl 0010.08104 · doi:10.1073/pnas.20.3.197
[37] Marshall Stone, ”Subsumption of the theory of Boolean algebras under the theory of Boolean rings,”Proceedings of the National Academy of Sciences 21, 1935, 103–105. · Zbl 0011.05104 · doi:10.1073/pnas.21.2.103
[38] Marshall Stone, ”The theory of representations for Boolean algebras,”Transactions of the American Mathematical Society 40, 1936, 37–111. · Zbl 0014.34002
[39] Marshall Stone, ”Applications of the theory of Boolean rings to general topology,”Transactions of the American Mathematical Society 41, 1937, 375–481. · Zbl 0017.13502 · doi:10.1090/S0002-9947-1937-1501905-7
[40] Alfred Tarski, ”Przyczynek do aksjomatyki zbioru dobrze uporzadkowanego,”Przeglad Filozoficzny (=Revue Philosophique) 24, 1921, 85–94.
[41] Alfred Tarski, ”O wyrazie pierwontnym logistyki,”Przeglad Filozoficzny (=Revue Philosophique) 26, 1923, 68–89. (A French translation appeared in the two articles ”Sur le terme primitif de la logistique,”Fundamenta Mathematicae 4, 1923, 196–200, and ”Sur les truth-functions au sense de MM. Russell et Whitehead,”Fundamenta Mathematicae 5, 1924, 59–74. An English translation appeared as Article I in [66].)
[42] Alfred Tarski, ”Sur quelques théorèmes qui équivalent à l’axiome du choix,”Fundamenta Mathematicae 2, 1924, 47–60.
[43] Alfred Tarski, ”Sur les ensembles finis,”Fundamenta Mathematicae 6, 1924, 45–95. · JFM 50.0135.02
[44] Alfred Tarski, ”O röwnowaznosci wielokatów,”Przeglad Matematyczno-fizyczny 2, 1924, 47–60.
[45] Alfred Tarski, ”Geschichtliche Entwicklung und gegenwärtiger Zustand der Gleichmächtigkeitstheorie und der Kardinalzahlarithmetik,” in:Ksiega Pamiatkowa Pierwszego Polskiego Zjazdu Matematycznego, Kraków, 1929, pp. 48–54.
[46] Alfred Tarski, ”Na marginesie ’Rozporzadzenia Prezydenta Rzeczypospolitej o ubezpieczeniu procownikow umyswych z dnia 24 listopada 1927 r’,”Ekonomista 29, 1929, 115–119.
[47] Alfred Tarski, ”Une contribution à la théorie de la mesure,”Fundamenta Mathematicae 15, 1930, 42–50. · JFM 56.0089.05
[48] Alfred Tarski, ”Über einige fundamentalen Begriffe der Metamathematik,”Sprawozdania z Posiedzeń Towarzystwa Naukowego Warszawskiego, WydziaIII Nauk Matematycznofizyznych (=Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III) 23, 1930, 22–29. (An English translation appeared as Article III in [66].)
[49] Alfred Tarski, ”Fundamentale Begriffe der Methodologie der deduktiven Wissenschaften. I,”Monatshefte für Mathematik und Physik 37, 1930, 361–404. (An English translation appeared as Article V in [66].) · JFM 56.0046.02 · doi:10.1007/BF01696782
[50] Alfred Tarski, ”Sur les ensembles définissables de nombres réels. I,”Fundamenta Mathematicae 17, 1931, 210–239. (An English translation appeared as Article VI in [66].) · JFM 57.0060.02
[51] Alfred Tarski, ”Zur Grundlegung der Boole’schen Algebra. I,”Fundamenta Mathematicae 24, 1935, 177–198. (An English translation appeared as Article XI in [66].) · JFM 61.0054.02
[52] Alfred Tarski, ”Grundzüge des Systemenkalküls. Erster Teil,”Fundamenta Mathematicae 25,1935, 503-526. (An English translation appeared as Article XII in [66].) · Zbl 0012.38501
[53] Alfred Tarski, ”Grundzüge des Systemenkalküls. Zweiter Teil,”Fundamenta Mathematicae 26,1936, 283-302. (An English translation appeared as Article XII in [66].) · JFM 62.0038.02
[54] Alfred Tarski, ”Der Wahrheitsbegriff in den formalisierten Sprachen,”Studia Philosophica 1, 1935, 261–405. (This is aGerman translation of ”Pojęcie prawdy w językach nauk dedukcyjnych,Prace Towarzystwa Naukowego Warszawskiego, WydziaII Nauk Matematyczno-fizycznych (= Travaux de la Société des Sciences et des Lettres de Varsovie, Classe III Sciences Mathématiques et Physiques), no. 34, Warsaw, 1933. An English translation appeared as Article VIII in [66].)
[55] Alfred Tarski, ”Einige methodologische Untersuchungen über die Definierbarkeit der Begriffe,”Erkenntnis 5, 1935, 80–100. (This is a German translation of ”Z badan metodologicznych nad definjowalnoscia terminöw,” PrzeglądFilozoficzny (=Revue Philosophique) 37, 1934, 438–460. An English translation appeared as Article X in [66].) · Zbl 0012.00101 · doi:10.1007/BF00172286
[56] Alfred Tarski, ”Über den Begriff der logischen Folgerung,” in:Actes du Congrès International de Philosophie Scientifique, vol. 7, 1936, pp. 1–11. (This is a German translation of ”O pojęciu wynikania logicznego,” PrzeglądFilozoficzny (=Revue Philosophique) 39, 1936, 58–68. An English translation appeared as Article XVI in [66].)
[57] Alfred Tarski, ”Ideale in den Mengenkörpern,”Rocznik Polskiego Towarzystwa Matematycznego (=Annales de la Société Polonaise de Mathématique) 15, 1936, 186–189.
[58] Alfred Tarski, ”On the calculus of relations,”Journal of Symbolic Logic 6, 1941, 73–89. · Zbl 0026.24401 · doi:10.2307/2268577
[59] Alfred Tarski, ”The semantic conception of truth and the foundations of semantics,”Philosophy and Phenomenological Research 4, 1944, 341–376. · Zbl 0061.00807 · doi:10.2307/2102968
[60] Alfred Tarski,A decision method for elementary algebra and geometry (prepared for publication by J. C. C. McKinsey), Santa Monica CA: RAND Corporation, 1948. (Revised edition: Berkeley: University of California Press, 1951.) · Zbl 0035.00602
[61] Alfred Tarski, ”On representable relation algebras,”Bulletin of the American Mathematical Society 58, 1952, 172. · Zbl 0049.00702
[62] Alfred Tarski, ”A formalization of set theory without variables,”Journal of Symbolic Logic 18, 1953, 189. · Zbl 0654.03036
[63] Alfred Tarski, ”Contributions to the theory of models. I,”Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series A, 57 (=Indagationes Mathematicae 16), 1954, 572–581. · Zbl 0058.24702
[64] Alfred Tarski, ”Contributions to the theory of models. II,”Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series A, 57 (=Indagationes Mathematicae 16), 1954, 582–588. · Zbl 0058.24702
[65] Alfred Tarski, ”Contributions to the theory of models. III,”Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series A, 58 (=Indagationes Mathematicae 17), 1955, 56–64. · Zbl 0058.24702
[66] Alfred Tarski,Logic, semantics, metamathematics, Oxford: Clarendon Press, 1956.
[67] Alfred Tarski,The completeness of elementary algebra and geometry, Paris: Institut Blaise Pascal, 1967.
[68] Alfred Tarski and Steven Givant,A formalization of set theory without variables, Providence, Rl: American Mathematical Society, 1987. · Zbl 0654.03036
[69] Jean van Heijenoort,From Frege to Gödel. A source book in mathematical logic, 1879–1931, Cambridge, MA: Harvard University Press, 1967. · Zbl 0183.00601
[70] Hermann Weyl, ”Über die Definitionen der mathematischen Grundbegriffe,”Mathematisch-naturwissenschaftliche Blätter 7, 1910, 93–95, 109–113. · JFM 41.0089.03
[71] Ernst Zermelo, ”Beweis, daß jede Menge wohlgeordnet werden kann,”Mathematische Annalen 59, 1904, 514–516. · JFM 35.0088.03 · doi:10.1007/BF01445300
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