Abstract
I want to tell you something about the personal and scientific relationship between Alfred Tarski and Kurt Gödel, more or less chronologically. This is part of a work in progress with Anita Feferman on a biography of Alfred Tarski, and in line with most of the things we do, we’ve talked a great deal about the subject together.
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Notes
The basic references for these developments are: Jan Woletíski, Logic and Philosophy in the Lvov-Warsaw School. Dordrecht: Kluwer 1989, and Kazimierz Kuratowski, A Half Century of Polish Mathematics. Oxford: Pergamon Press 1980.
Besides Lesniewski and Lukasiewicz, among Tarski’s professors at the University of Warsaw were the philosopher Tadeusz Kotarbinski and the mathematicians Kazimierz Kuratowski, Stefan Mazurkiewicz and Waclaw Sierpiúski.
One that he used forever after - personally I find it rather heavy in character, but it was clearly to his taste.
Tarski’s publications (except for one posthumous piece), are reproduced from the originals in: Alfred Tarski, Collected Papers, Vols. 1–4. Basel/Boston: Birkhäuser 1986.
The primarily logical papers from 1923–1938 are to be found in English translation by J.H.Woodger in Alfred Tarski, Logic, Semantics, Metamathematics. ( Second Edition) Indianapolis: Hackett 1983.
Informative surveys of the full body of Tarski’s work are given in a series of articles by various authors in The Journal of Symbolic Logic 51, 1986, pp.866–941 and ibid. 53, 1988, pp.291. Finally, the article by Steven Givant, “A Portrait of Alfred Tarski”, in: The Mathematical I telligencer 13, 1991, pp.16–32, contains much biographical information and a general picture of Tarski’s interests and working habits.
The circumstances of Tarski’s 1930 visit to Vienna are recounted in “Poland and the Vienna Circle”, Chapter XII in Karl Menger, Reminiscences of the Vienna Circle and the Mathematical Colloquium. Dordrecht: Kluwer 1994.
See Eckehart Köhler, “Gödel und der Wiener Kreis”, in: Paul Kruntorad (ed.), Jour Fixe der Vernun f t. Der Wiener Kreis und die Folgen,Wien: Hölder-Pichler-Tempsky, § 3.2.
Gödel’s writings, both published and unpublished, are referred to in the following via the volumes: Kurt Gödel, Collected Works, Vol.l. Publications 1929–1936. New York: Oxford University Press 1986, Kurt Gödel, Collected Works, Vol.!!. Publications 1938–1974. New York: Oxford University Press 1990, and Kurt Gödel, Collected Works, Vol.III. Unpublished Essays and Lectures. New York: Oxford University Press 1995. The form of reference to individual pieces within these volumes (where they appear both in the original language and in English translation) is Gödel 19xx for published work and Gödel *19xx for unpublished writings and lectures. We shall follow that scheme below. Thus, for example, Gödel’s dissertation is Gödel 1929.
As general sources for Gödel’s life and work drawn upon in the following, we make use of: John W. Dawson, Jr., Logical Dilemmas. The Life and Work of Kurt Gödel. Wellesley: A.K.Peters, Ltd. 1997, and Solomon Feferman, “The Life and Work of Kurt Gödel”, in the Gödel Collected Works, Vol.I,pp.1–36.
To appear along with several other items of correspondence between Tarski and Gödel in the Gödel Collected Works, Vol. IV (in progress).
Presumably) Alfred Tarski, “Fundamentale Begriffe der Methodologie der deduktiven Wissenschaften. I”, in: Monatshefte fir Mathematik und Physik 37, 1930, pp.361–404.
The rights to Gödel’s scientific Nachlass reside with the Institute for Advanced Study in Princeton; it is on indefinite loan to the Manuscripts Division of the Firestone Library at Princeton University.
Thanks to Jan Tarski, his father’s side of this correspondence is presented in its entirety in this volume. Most of these are of a personal nature and will not be included in the volume mentioned in footnote 7. Only the letters dated 24 March 1944 and 10 December 1946 have been deemed by the editors of that volume to be of sufficient scientific interest to warrant inclusion there. On Gödel’s side there are just three such items, including that of 20 January 1931 reproduced in the text, 12 April 1944, and (?) August 1961.
This has appeared as Gödel *1970c in Vol. III of the Collected Works.
From the translation of the Wahrheitsbegriff in: Logic, Semantics, Metamathematics,pp.277–278.
Solomon Feferman, “Kurt Gödel: Conviction and Caution”, in: Philosophia Naturalis 21, 1984, pp.546–562.
See the introductory note to Gödel 1934 in Vol. I of the Collected Works.
From the translation of Tarski’s 1930 paper in Logic, Semantics, Metamathematics, pp.93–94. There is a footnote to the quoted passage citing Abraham Fraenkel, Einleitung in die Mengenlehre, 3rd. ed. Berlin: Springer, pp. 347–354.
Cf. Jean van Heijenoort, A Source Book in Mathematical Logic. Cambridge: Harvard Univ. Press, 1967, p.285, and Hermann Weyl, “Über die Definitionen der mathematischen Grundbegriffe”, in: Mathematisch-naturwissenschaftliche Blätter 7, 1910, pp.93–95 and pp.109–113.
The photo is reproduced in the Gödel Collected Works, Vol. I,between pp.15 and 16. It came from the Tarski archives in the Bancroft Library of the University of California, Berkeley.
See the introductory note by Robert Solovay to Gödel 1938, 1939 and 1940 in Vol. II of the Collected Works. The relationship between Gödel’s constructible hierarchy and Russell’s ramified hierarchy (as well as Hilbert’s abortive “proof” of the Continuum Hypothesis in 1928) is discussed in Solovay’s introductory note to the previously unpublished lecture Gödel *1938 to be found in Vol. III of the Collected Works.
Jeff Paris and Leo Harrington, “A Mathematical Incompleteness in Peano Arithmetic”, in: Jon Barwise, Ed., Handbook of Mathematical Logic. Amsterdam: North Holland, 1977, pp.11331142. See also Solomon Feferman, “Does Mathematics Need New Axioms?”, American Mathematical Monthly (to appear).
See Donald A. Martin, “Hilbert’s First Problem: The Continuum Hypothesis”, in: Felix Browder, Ed., Mathematical Developments Arising from Hilbert Problems. Providence: Amer. Math. Soc., 1976, pp.81–92. See also Feferman, ibid.
John Myhill and Dana S. Scott, “Ordinal Definability”, in: Dana S. Scott, Ed., Axiomatic Set Theory. Providence: Amer. Math. Soc., 1971, pp.271–278. See also the introductory note by Charles Parsons to Gödel 1946 in Vol. II of the Collected Works.
This photo was also taken by Maria Kokoszynska-Lutman. Its source is Gödel’s Nachlass,and is to be found reproduced on p.252 of the Gödel Collected Works, Vol. II.
Cf. Georg Kreisel and Azriel Levy, “Reflection Principles and Their Use for Establishing the Complexity of Axiomatic Systems, in: Zeitschr. für mathematische Logik u. Grundlagen der Mathematik 14, pp.97–142. Further applications of partial truth definitions as a basic method are presented in Peter Hâjek and Pavel Pudlâk, Metamathematics of First-order Arithmetic. Berlin: Springer, 1993.
The circumstances and technical aspects of this are discussed in full in the introductory note to Gödel *1970a, *19706 and *1970c by Robert Solovay in Vol. III of the Collected Works. The unsent letter itself is given as *1970c, op. cit., pp.424–425.
Namely in his use of König’s Infinity Lemma in the proof of the completeness of the predicate calculus; cf. the introductory note by Burton Dreben and Jean van Heijenoort to Gödel 1929, 1930 and 1930a in Vol. I of the Collected Works.
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Feferman, S. (1999). Tarski and Gödel: Between the Lines. In: Woleński, J., Köhler, E. (eds) Alfred Tarski and the Vienna Circle. Vienna Circle Institute Yearbook [1998], vol 6. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0689-6_5
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