Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-05T21:09:34.339Z Has data issue: false hasContentIssue false
  • English
  • Français

Address at the Princeton University Bicentennial Conference on Problems of Mathematics (December 17–19, 1946), By Alfred Tarski

Published online by Cambridge University Press:  15 January 2014

Hourya Sinaceur
Hourya Sinaceur*
Affiliation:
Cnrs-Paris I, Ihpst, 13 Rue Du Four, 75006 Paris, FranceE-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This article presents Tarski's Address at the Princeton Bicentennial Conference on Problems of Mathematics, together with a separate summary. Two accounts of the discussion which followed are also included. The central topic of the Address and of the discussion is decision problems. The introductory note gives information about the Conference, about the background of the subjects discussed in the Address, and about subsequent developments to these subjects.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 6 , Issue 1 , March 2000 , pp. 1 - 44
Copyright
Copyright © Association for Symbolic Logic 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1947] Problems of mathematics, Princeton University bicentennial conferences, series 2, conference 2, Princeton, New Jersey, 1947, reprint in [Duren, 1989, pp. 309–334].Google Scholar
[1954] Ackermann, Wilhelm, Solvable cases of the decision problem, North-Holland, Amsterdam, 1954.Google Scholar
[1954] Addison, John W. Jr., On some points of the theory of recursive functions, Doctoral dissertation, University of Wisconsin, 1954, iv+92 pages.Google Scholar
[1955] Addison, John W., Analogies in the Borel, Lusin, and Kleene hierarchies, I and II, Bulletin of the American Mathematical Society, vol. 61 (1955), pp. 75 and 171–172.Google Scholar
[1959] Addison, John W., Separation principles in the hierarchies of classical and effective descriptive set theory, Fundamenta Mathematicae, vol. 46 (1959), pp. 123–135.CrossRefGoogle Scholar
[1965] Barnes, R. F. Jr., The classification of recursive sets of number-theoretic functions, Notices of the American Mathematical Society, vol. 12 (1965), pp. 622–623.Google Scholar
[1922] Behmann, Heinrich, Beiträge zur Algebra der Logik, insbesondere zum Entscheidungsproblem, Mathematische Annalen, vol. 86 (1922), pp. 163–229.CrossRefGoogle Scholar
[1936] Birkhoff, Garett and Neumann, John von, The logic of quantum mechanics, Annals of Mathematics, vol. 37 (1936), pp. 823–843.CrossRefGoogle Scholar
[1961] Bocheński, Innocent Marie, A history of formal logic, Notre Dame, Indiana, 1961, english translation of Formale Logik , Verlag Karl Alberg, Freiburg/München, 1956.Google Scholar
[1936a] Church, Alonzo, An unsolvable problem of elementary number theory, The American Journal of Mathematics, vol. 58 (1936), pp. 345–363, reprint in [Davis, 1965, pp. 88–107].CrossRefGoogle Scholar
[1936b] Church, Alonzo, A note on the entcheidungsproblem, The Journal of Symbolic Logic, vol. 1 (1936), pp. 40–41, correction ibid., pp. 101–102, reprint in [Davis, 1965, pp. 108–115].CrossRefGoogle Scholar
[1956] Church, Alonzo, Introduction to mathematical logic, vol. I, Princeton University Press, 1956.Google Scholar
[1958] Davis, Martin, Computability and unsolvability, McGraw-Hill series in information processing and computers, McGraw-Hill Book Company, New York-Toronto-London, 1958.Google Scholar
[1965] Davis, Martin (editor), The undecidable: Basic papers on undecidable propositions, unsolvable problems and computable functions, Raven Press, Hewlett, N.Y., 1965.Google Scholar
[1976] Davis, Martin, Matijasevich, Yuri, and Robinson, Julia, Hilbert's tenth problem. Diophantine equations: positive aspects of a negative solution, Proceedings of symposia in pure mathematics (Browder, F., editor), no. 28, 1976, pp. 323–378.Google Scholar
[1961] Davis, Martin, Putnam, Hilary, and Robinson, Julia, The undecidability of exponential Diophantine equations, Annals of Mathematics, vol. 74 (1961), no. 2, pp. 425–436.CrossRefGoogle Scholar
[1988] Doner, John and Hodges, Wilfrid, Alfred Tarski and decidable theories, The Journal of Symbolic Logic, vol. 53 (1988), no. 1, pp. 20–35.CrossRefGoogle Scholar
[1989] Duren, Peter (editor), A century of mathematics in America. History of mathematics, vol. 2, American Mathematical Society, Providence, Rhode Island, 1989, pp. 309–334.Google Scholar
[1962] Feferman, Solomon, Transfinite recursive progressions of axiomatic theories, The Journal of Symbolic Logic, vol. 27 (1962), pp. 259–316.CrossRefGoogle Scholar
[1988] Feferman, Solomon, Turing in the land of O(z), The universal Turing machine: A half-century survey (Herken, Rolf, editor), Clarendon Press, Oxford, 1988, pp. 113–147. Google Scholar
[1999] Feferman, Solomon, Tarski and Gödel: Between the lines, 1999, in [Woleński and Köhler, 1999, pp. 53–63].CrossRefGoogle Scholar
[1959] Feferman, Solomon and Vaught, Robert, The first order properties of products of algebraic systems, Fundamenta Mathematicae, vol. 47 (1959), pp. 57–103.CrossRefGoogle Scholar
[1974] Fischer, M. J. and Rabin, M. O., Super-exponential complexity of Presburger arithmetic, Complexity of computation, SIAM-AMS proceedings (Karp, R. M., editor), vol. 7, 1974, pp. 27–41.Google Scholar
[1991] Givant, Stephen R., A portrait of Alfred Tarski, The Mathematical Intelligencer, vol. 13 (1991), no. 3, pp. 16–31.CrossRefGoogle Scholar
[1986–] Gödel, Kurt, Collected Works, edited by Feferman, Solomon, Dawson, John W. Jr., Kleene, Stephen C., Moore, Gregory H., Solovay, Robert and Heijenoort, Jean van, Oxford University Press, vol. I, 1986; vol. II, 1990; vol. III ed. by Solomon Feferman, John W. Dawson Jr., Warren Goldfarb, Charles Parsons and Robert Solovay, 1995, 1986.Google Scholar
[1967] Heinjenoort, Jean van, From Frege to Gödel. A source book in mathematical logic 1879–1931, Harvard University Press, Cambridge, Massachusetts, 1967.Google Scholar
[1922] Hilbert, David, Neubegründung der Mathematik. Erste Mitteilung, Abhandlungenaus dem mathematischen Seminar der Hamburgischen Universität 1 (1922), in [Hilbert, 1935, pp. 157–177].Google Scholar
[1935] Hilbert, David, Gesammelte Abhandlungen, vol. III, Springer, Berlin, 1935, reprint: Chelsea, New York, 1965.Google Scholar
[1928] Hilbert, David and Ackermann, Wilhelm, Grundzüge der theoretischen Logik, Springer, Berlin, 1928.Google Scholar
[1974, 1980] Jacobson, Nathan, Basic algebra, vol. I, II, W. H. Freeman & Co, San Francisco, 1974, 1980.Google Scholar
[1943] Kleene, Stephen C., Recursive predicates and quantifiers, Transactions of the American Mathematical Society, vol. 53 (1943), pp. 41–73, reprint in [Davis, 1965, pp. 254–287].CrossRefGoogle Scholar
[1950] Kleene, Stephen C., A symmetric form of Gödel's theorem, Indagationes Mathematicae, vol. 12 (1950), pp. 244–246.Google Scholar
[1958] Kleene, Stephen C., Extension of an effectively generated class of functions by enumeration, Colloquium Mathematicum, vol. 6 (1958), pp. 67–78.CrossRefGoogle Scholar
[1887] Kronecker, Leopold, Über den Zahlbegriff, Journal für die reine und angewandte Mathematik, vol. 101 (1887), pp. 337–355.CrossRefGoogle Scholar
[1927] Langford, Cooper Harold, Some theorems on deducibility, Annals of Mathematics, vol. 28 (1927), pp. 16–40 and 459–471.CrossRefGoogle Scholar
[1932] Lewis, Clarence Irwing and Langford, Cooper Harold, Symbolic logic, The Century Co, New York, 1932.Google Scholar
[1949] Linial, Samuel and Post, Emil, Recursive unsolvability of the deducibility, Tarski's completeness and independence of axioms problems of the propositional calculus, Bulletin of the American Mathematical Society, vol. 55 (1949), p. 50, abstract.Google Scholar
[1915] Löwenheim, Leopold, Über Möglichkeiten im Relativkalkül, Mathematische Annalen, vol. 76 (1915), pp. 447–470, english translation in [van Heinjenoort, 1967, pp. 228–251].CrossRefGoogle Scholar
[1930] Łukasiewicz, Jan and Tarski, Alfred, Untersuchungen über den Aussagenkalkül, Comptes Rendus de la Société des Sciences et des Lettres de Varsovie, vol. XXIII, Cl. 3 (1930), pp. 30–50, english translation in [Tarski, 1956, pp. 38–59], in [Tarski, 1986a, vol. 1, pp. 321–343].Google Scholar
[1925] Luzin, Nicolas, Sur les ensembles projectifs de M. Henri Lebesgue, Comptes rendus hebdomadaires des séances de l' Académie des Sciences (Paris), vol. 180 (1925), pp. 1572–1574, and two related articles in the same volume, pp. 1318–1320 and 1817–1819.Google Scholar
[1996] Macintyre, Angus and Wilkie, Alex J., On the decidability of the real exponential field, Kreiseliana. About and around Georg Kreisel (Odifreddi, p., editor), A. K. Peters, Wellesley, Massachusetts, 1996, pp. 441–468.Google Scholar
[1941] McKinsey, J. C. C., A solution of the decision problem for the Lewis systems S2 and S4, with an application to topology, The Journal of Symbolic Logic, vol. 6 (1941), no. 1, pp. 117–134.CrossRefGoogle Scholar
[1980] Moschovakis, Yiannis N., Descriptive set theory, North Holland, Amsterdam, 1980.Google Scholar
[1947] Mostowski, Andrzej, On definable sets of positive integers, Fundamenta Mathematicae, vol. 34 (1947), pp. 81–112.CrossRefGoogle Scholar
[1949a] Mostowski, Andrzej and Tarski, Alfred, Arithmetical classes and types of well ordered systems, The Bulletin of the American Mathematical Society, vol. 55 (1949), p. 65, abstract, in [Tarski, 1986a, vol. 4, p. 583].Google Scholar
[1949b] Mostowski, Andrzej, Undecidability in the arithmetic of integers and in the theory of rings, The Journal of Symbolic Logic, vol. 14 (1949), no. 1, p. 76, in [Tarski, 1986a, vol. 4, p. 586].Google Scholar
[1977] Paris, Jeff and Harrington, Leo, A mathematical incompleteness in Peano arithmetic, Handbook of mathematical logic (Barwise, J., editor), North-Holland, Amsterdam, 1977, pp. 1133–1142.CrossRefGoogle Scholar
[1885] Peirce, Charles Sanders, On the algebra of logic: Contribution to the philosophy of notation, American Journal of Mathematics, vol. 7 (1885), pp. 180–202.CrossRefGoogle Scholar
[1921] Post, Emil, Introduction to a general theory of elementary propositions, American Journal of Mathematics, vol. 43 (1921), pp. 1163–185, reprint in [van Heinjenoort, 1967, pp. 264–283].CrossRefGoogle Scholar
[1944] Post, Emil, Recursively enumerable sets of positive integers and their decision problems, Bulletin of the American Mathematical Society, vol. 50 (1944), reprint in [Davis, 1965, pp. 305–337].CrossRefGoogle Scholar
[1930] Presburger, Mojżesz, Über die vollständigkeit eines gewissen systems der arithmetik ganzer zahlen, in welchem die addition als einzige operation hervortritt, Sprawozdanie z I kongresu matematików krajów slowianskich, (Compte rendus du I congrès des mathématiciens des pays slaves) (Warszawa 1929), 1930, pp. 92–101, 395.Google Scholar
[1985] Orman Quine, Willard van, The time of my life. An autobiography, A Bradford book, the MIT Press, Cambridge (Massachusetts)-London, 1985.Google Scholar
[1990] Reid, Constance, The autobiography of Julia Robinson, More mathematical people, Academic Press, 1990, reprint in [Reid, 1996], pp. 262–280.Google Scholar
[1996] Reid, Constance, Julia. A life in mathematics, The Mathematical Association of America, Washington DC, 1996.Google Scholar
[1949a] Robinson, Julia, Undecidability in the arithmetic of integers and rationals and in the theory of fields, The Journal of Symbolic Logic, vol. 14 (1949), no. 1, p. 77, abstract.Google Scholar
[1949b] Robinson, Julia, Definability and decision problems in arithmetic, The Journal of Symbolic Logic, vol. 14 (1949), no. 2, pp. 98–114.CrossRefGoogle Scholar
[1969] Robinson, Julia, Unsolvable Diophantine problems, Proceedings of the American Mathematical Society, vol. 22 (1969), pp. 534–538.CrossRefGoogle Scholar
[1985] Robinson, Julia, The collected works of Julia Robinson (Feferman, Solomon, editor), American Mathematical Society, Providence R. I., 1985.Google Scholar
[1989] Roddy, Michael S., On the word problem for orthocomplemented modular lattices, Canadian Journal of Mathematics, vol. 41 (1989), pp. 961–1004.CrossRefGoogle Scholar
[1936] Rosser, John Barkley, Extensions of some theorems of Gödel and Church, The Journal of Symbolic Logic, vol. 1 (1936), pp. 87–91, reprint in [Davis, 1965, pp. 231–235].CrossRefGoogle Scholar
[1950] Sierpiński, M. W., Les ensembles projectifs et analytiques, Gauthier-Villars, Paris, 1950.Google Scholar
[1991] Sinaceur, Hourya, Corps et modèles. Essai sur l' histoire de l' algèbre réelle, Vrin, Paris, 1991, an english version is to be published by Birkhäuser in 2000.Google Scholar
[1919] Skolem, Thoralf, Untersuchugen über die axiome des klassenkalküls une über produktions—und summationsprobleme, welche gewisse klassen von aussagen betreffen, Skrifter Videnskapsakademiet i Kristiana, vol. 3 (1919), pp. 37–71, reprint in [Skolem, 1970, pp. 67–102].Google Scholar
[1930] Skolem, Thoralf, Über einige satzfunktionen in der arithmetik, Skrifter Videnskapsakademiet i Oslo, vol. I (1930), no. 7, reprint in [Skolem, 1970, pp. 281–306].Google Scholar
[1970] Skolem, Thoralf, Selected works in logic (Fenstad, J. E., editor), Universitetsforlaget, Oslo, 1970.Google Scholar
[1971] Solovay, Robert and Tennenbaum, S., Iterated Cohen extensions and Souslin's problem, Annals of Mathematics, vol. 94 (1971), pp. 201–245.CrossRefGoogle Scholar
[1917] Suslin, Michael J., Sur une définition des ensembles mesurables B sans nombres transfinis, Comptes rendus hebdomadaires des séances de l' Académie des Sciences (Paris), vol. 164 (1917), pp. 88–91.Google Scholar
[1949] Szmielew, Wanda, Decision problem in group theory, Proceedings of the Xth international congress of philosophy (Amsterdam 1948), North-Holland, Amsterdam, 1949, pp. 763–766.Google Scholar
[1955] Szmielew, Wanda, Elementary properties of Abelian groups, Fundamenta Mathematicae, vol. 41 (1955), pp. 203–271, (abstract : Bulletin of the American Mathematical Society 55 (1949), p. 65).CrossRefGoogle Scholar
[1930a] Tarski, Alfred, Über einige fundamentale Begriffe der Metamathematik, Comptes Rendus de la Société des Sciences et des Lettres de Varsovie, vol. XXIII, Cl. 3 (1930), in [Tarski, 1986a, vol. 1, pp. 311–320].Google Scholar
[1930b] Tarski, Alfred, Fundamentale Begriffe der Methodologie der deduktiven Wissenschaften I, Monatshefte für Mathematik und Physik, vol. 37 (1930), pp. 361–404, in [Tarski, 1986a, vol. 1, pp. 341–390].CrossRefGoogle Scholar
[1931] Tarski, Alfred, Sur les ensembles définissables de nombres réels, I, Fundamenta Mathematicae, vol. 17 (1931), pp. 210–239, in [Tarski, 1986a, vol. 1, pp. 517–548].CrossRefGoogle Scholar
[1935] Tarski, Alfred, Grundzüge des Systemenkalküls, Zweiter Teil, Fundamenta Mathematicae, vol. 25 (1935), pp. 503–526, in [Tarski, 1986a, vol. 2, pp. 25–50].CrossRefGoogle Scholar
[1936] Tarski, Alfred, Grundzüge des Systemenkalküls, zweiter Teil, Fundamenta Mathematicae, vol. 26 (1936), pp. 283–301, in [Tarski, 1986a, vol. 2, pp. 223–244].CrossRefGoogle Scholar
[1939/1967] Tarski, Alfred, The completeness of elementary algebra and geometry, Centre National de la Recherche Scientifique, Institut Blaise Pascal, Paris, 1967, in [Tarski, 1986a, vol. 4, pp. 289–346].Google Scholar
[1939] Tarski, Alfred, On undecidable statements in enlarged systems of logic and the concept of truth, The Journal of Symbolic Logic, vol. 4 (1939), no. 3, pp. 105–112, in [Tarski, 1986a, vol. 2, pp. 559–568].CrossRefGoogle Scholar
[1948] Tarski, Alfred, A decision method for elementary algebra and geometry, 1948, (prepared for publication by J. C. McKinsey). Second revised edition, University of California Press, 1951. In [Tarski, 1986a, vol. 3, pp. 297–368].CrossRefGoogle Scholar
[1949a] Tarski, Alfred, On essential undecidability, The Journal of Symbolic Logic, vol. 14 (1949), no. 1, pp. 75–76, abstract, in [Tarski, 1986a, vol. 4, pp. 584–585].Google Scholar
[1949b] Tarski, Alfred, Undecidability of group theory, The Journal of Symbolic Logic, vol. 14 (1949), no. 1, pp. 76–77, abstract, in [Tarski, 1986a, vol. 4, pp. 588–589].Google Scholar
[1949c] Tarski, Alfred, Undecidability of the theories of lattices and projective geometries, The Journal of Symbolic Logic, vol. 14 (1949), no. 1, pp. 77–78, abstract, in [Tarski, 1986a, vol. 4, pp. 590–591].Google Scholar
[1952] Tarski, Alfred, Some notions on the borderline of algebra and metamathematics, Proceedings of the international congress of mathematicians (Cambridge, Massachusetts, 1950), 1952, in [Tarski, 1986a, vol. 3, pp. 459–476].Google Scholar
[1956] Tarski, Alfred, Logic, semantics, metamathematics. Papers from 1923 to 1938, Clarendon Press, Oxford, 1956, translated by J. H.Woodger. Second revised edition with an introduction by John Corcoran, Hackett Publishing Company, Indianapolis, Indiana, 1983.Google Scholar
[1986a] Tarski, Alfred, Collected papers (Givant, S. R. and McKenzie, R. N., editors), Birkhäuser, 1986.Google Scholar
[1986b] Tarski, Alfred, What are logical notions?, History and Philosophy of Logic, vol. 7 (1986), pp. 143–154, edited by Corcoran, J..CrossRefGoogle Scholar
[1953] Tarski, Alfred, Mostowski, Andrzej, and Robinson, Raphael M., Undecidable theories, North-Holland, Amsterdam, 1953.Google Scholar
[1957] Tarski, Alfred and Vaught, Robert, Arithmetical extensions of relational systems, Compositio Mathematica, vol. 13 (1957), pp. 81–102, in [Tarski, 1986a, vol. 3, pp. 651–674]. Google Scholar
[1936] Turing, Alan, On computable numbers, with an application to the entscheidungsproblem, Proceedings of the London Mathematical Society, ser. 2, vol. 42 (1936–1937), pp. 230–265, corrections ibid., vol. 43 (1937), pp. 544–546, reprint in [Davis, 1965, pp. 115–154].CrossRefGoogle Scholar
[1939] Turing, Alan, Systems of logic based on ordinals, Proceedings of the London Mathematical Society, ser. 2, vol. 45 (1939), pp. 161–228, reprint in [Davis, 1965, pp. 154–222].CrossRefGoogle Scholar
[1974] Vaught, Robert, Model theory before 1945, Proceedings of the Tarski symposium, Proceedings of symposia in pure mathematics, American Mathematical Society, Providence, Rhode Island, 1974, pp. 154–172.Google Scholar
[1996] Wilkie, Alex J., Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, Journal of the American Mathematical Society, vol. 9 (1996), pp. 1051–1094.CrossRefGoogle Scholar
[1995] Woleński, Jan, On Tarski's background, From Dedekind to Gödel. Essays on the development of the foundations of mathematics (Hintikka, Jaakko, editor), Kluwer Academic Publishers, 1995, pp. 331–342.CrossRefGoogle Scholar
[1999] Woleński, Jan and Köhler, Eckehart (editors), Alfred Tarski and the Vienna Circle. Austro-Polish connections in logical empiricism, Kluwer Academic Publishers, 1999.CrossRefGoogle Scholar