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Gödel and Set Theory

Published online by Cambridge University Press:  15 January 2014

Akihiro Kanamori
Akihiro Kanamori*
Affiliation:
Department of Mathematics, Boston University, Boston, Massachusetts 02215, USAE-mail: aki@math.bu.edu
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Extract

Kurt Gödel (1906–1978) with his work on the constructible universe L established the relative consistency of the Axiom of Choice (AC) and the Continuum Hypothesis (CH). More broadly, he ensured the ascendancy of first-order logic as the framework and a matter of method for set theory and secured the cumulative hierarchy view of the universe of sets. Gödel thereby transformed set theory and launched it with structured subject matter and specific methods of proof. In later years Gödel worked on a variety of set theoretic constructions and speculated about how problems might be settled with new axioms. We here chronicle this development from the point of view of the evolution of set theory as a field of mathematics. Much has been written, of course, about Gödel's work in set theory, from textbook expositions to the introductory notes to his collected papers. The present account presents an integrated view of the historical and mathematical development as supported by his recently published lectures and correspondence. Beyond the surface of things we delve deeper into the mathematics. What emerges are the roots and anticipations in work of Russell and Hilbert, and most prominently the sustained motif of truth as formalizable in the “next higher system”. We especially work at bringing out how transforming Gödel's work was for set theory. It is difficult now to see what conceptual and technical distance Gödel had to cover and how dramatic his re-orientation of set theory was.

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Bulletin of Symbolic Logic , Volume 13 , Issue 2 , June 2007 , pp. 153 - 188
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Copyright © Association for Symbolic Logic 2007

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References

REFERENCES

Bekkali, Mohamed [1991], Topics in set theory, Lecture Notes in Mathematics, no. 1476, Springer-Verlag, New York.Google Scholar
Bernays, Paul [1937], A system of axiomatic set theory. Part I, The Journal of Symbolic Logic, vol. 2, pp. 65–77, reprinted in: Gert H. Müller (editor), Sets and classes , North-Holland, Amsterdam, pp. 1–13.Google Scholar
Brendle, Jörg, Larson, Paul, and Todorčević, Stevo [∞], Rectangular axioms, perfect set properties and decomposition, to appear.Google Scholar
Davis, Martin (editor) [1965], The undecidable: Basic papers on undecidable propositions, unsolvable problems and computable functions, Raven Press, Hewlett, New York.Google Scholar
Dawson, John W. Jr. [1997], Logical dilemmas: The life and work of Kurt Gödel, A K Peters, Wellesley.Google Scholar
Du Bois–Reymond, Paul [1880], Der Beweis des Fundamentalsatzes der Integralrechnung: F′(x) dx = F(b) − F(a), Mathematisches Annalen, vol. 16, pp. 115–128.Google Scholar
Du Bois–Reymond, Paul [1882], Die allgemeine Funktionentheorie I, Lampp, Tübingen.Google Scholar
Feferman, Solomon [1984], Kurt Gödel: Conviction and caution, Philosophia Naturalis, vol. 21, pp. 546–562, reprinted in [1998] below, pp. 150–164.Google Scholar
Feferman, Solomon (editor) [1986], Kurt Gödel, Collected works, Publications 1929–1936, vol. I, Oxford University Press, New York.Google Scholar
Feferman, Solomon [1987], Infinity in mathematics: Is Cantor necessary? (Conclusion), (di Francia, G. Toraldo, editor), L'infinito nella Scienza, Instituto della Enciclopedia Italiana, Roma, reprinted in [1998] below, particularly pp. 229–248, pp. 151–209.Google Scholar
Feferman, Solomon (editor) [1990], Kurt Gödel, Collected works, Publications 1938–1974, vol. II, Oxford University Press, New York.Google Scholar
Feferman, Solomon (editor) [1995], Kurt Gödel, Collected works, Unpublished essays and lectures, vol. III, Oxford University Press, New York.Google Scholar
Feferman, Solomon [1998], In the light of logic, Oxford University Press, New York.CrossRefGoogle Scholar
Feferman, Solomon and Dawson, John W. Jr. (editors) [2003a], Kurt Gödel, Collected works, Correspondence A–G, vol. IV, Clarendon Press, Oxford.Google Scholar
Feferman, Solomon and Dawson, John W. Jr. (editors) [2003b], Kurt Gödel, Collected works, Correspondence H-Z, vol. V, Clarendon Press, Oxford.Google Scholar
Fenstad, Jens E. (editor) [1970], Thoralf Skolem, Selected works in logic, Universitetsfor-laget, Oslo.Google Scholar
Floyd, Juliet and Kanamori, Akihiro [2006], How Gödel transformed set theory, Notices of the American Mathematical Society, vol. 53, pp. 417–425.Google Scholar
Fraenkel, Abraham [1922], Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre, Mathematische Annalen, vol. 86, pp. 230–237.Google Scholar
Fraenkel, Abraham [1953], Abstract set theory, North Holland, Amsterdam.Google Scholar
Gödel, Kurt [1929], Über die Vollständigkeit der Logikkalküls, doctoral dissertation, University of Vienna, reprinted and translated in Feferman, [1986], pp. 60–101.Google Scholar
Gödel, Kurt [193?], Untitled lecture, in Feferman, [1995], pp. 164–175.Google Scholar
Gödel, Kurt [1931], Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik, vol. 38, pp. 173–198, reprinted and translated with minor emendations by the author in Feferman [1986], pp. 144–195.Google Scholar
Gödel, Kurt [1932], Über Vollständigkeit und Widerspruchsfreiheit, Ergebnisse eines mathematischen Kolloquiums, vol. 3, pp. 12–13, text and translation in Feferman [1995], pp. 234–237.Google Scholar
Gödel, Kurt [1933], The present situation in the foundations of mathematics, handwritten text for an invited lecture, in Feferman, [1995], pp. 45–53, and the page references are to these.Google Scholar
Gödel, Kurt [1938], The consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis, Proceedings of the National Academy of Sciences U.S.A., vol. 24, pp. 556–557, reprinted in Feferman [1990], pp. 26–27.Google Scholar
Gödel, Kurt [1939a], The consistency of the generalized continuum hypothesis, Bulletin of the American Mathematical Society, vol. 45, p. 93, reprinted in Feferman [1990], p. 27.Google Scholar
Gödel, Kurt [1939b], Consistency-proof for the generalized continuum-hypothesis, Proceedings of the National Academy of Sciences U.S.A., vol. 25, pp. 220–224, reprinted in Feferman [1990], pp. 28–32.Google Scholar
Gödel, Kurt [1939c], Vortrag Göttingen, text and translation in Feferman [1995], pp. 126–155, and the page references are to these.Google Scholar
Gödel, Kurt [1940a], The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory, Annals of Mathematics Studies, no. 3, Princeton University Press, Princeton, reprinted in Feferman, [1990], pp. 33–101.Google Scholar
Gödel, Kurt [1940b], Lecture [on the] consistency [of the] continuum hypothesis, (Brown University) in Feferman, [1995], pp. 175–185, and the page references are to these.Google Scholar
Gödel, Kurt [1944], Russell's mathematical logic, The philosophy of Bertrand Russell (Schilpp, Paul A., editor), The Library of Living Philosophers, vol. 5, Northwestern University, Evanston, reprinted in Feferman, [1990], pp. 119–141, pp. 123–153.Google Scholar
Gödel, Kurt [1947], What is Cantor's Continuum Problem?, American Mathematical Monthly, vol. 54, pp. 515–525, errata vol. 55 (1948), p. 151; reprinted in Feferman [1990], pp. 176–187; see also Gödel [1964].Google Scholar
Gödel, Kurt [1964], Philosophy of mathematics. Selected readings, (Benacerraf, Paul and Putnam, Hilary, editors), Prentice Hall, Englewood Cliffs, revised and expanded version of [1947]; this version reprinted with emendations by the author in Feferman [1990], pp. 254–270, pp. 258–273.Google Scholar
Gödel, Kurt [1970a], Some considerations leading to the probable conclusion that the true power of the continuum is ℵ2, handwritten document, in Feferman, [1995], pp. 420–422.Google Scholar
Gödel, Kurt [1970b], A proof of Cantor's continuum hypothesis from a highly plausible axiom about orders of growth, handwritten document, in Feferman, [1995], pp. 422–423.Google Scholar
Gödel, Kurt [1972], Some remarks on the undecidability results, in Feferman, [1990], pp. 305–306.Google Scholar
Hartmanis, Juris [1989], Gödel, von Neumann, and the P = NP problem, Bulletin of the European Association for Theoretical Computer Science, vol. 38, pp. 101–107.Google Scholar
Hausdorff, Felix [1907], Untersuchungen über Ordnungstypen, IV, V, Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse, vol. 59, pp. 84–159, translated in Plotkin [2005], pp. 113–171.Google Scholar
Hausdorff, Felix [1909], Die Graduierung nach dem Endverlauf, Abhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse, vol. 31, pp. 295–334, translated in Plotkin [2005], pp. 271–301.Google Scholar
Hausdorff, Felix [1936], Summen von ℵ1 Mengen, Fundamenta Mathematicae, vol. 26, pp. 241–255, translated in Plotkin [2005], pp. 305–316.Google Scholar
Hauser, Kai [2006], Gödel's program revisited, part I: The turn to phenomenology, this Bulletin, vol. 12, pp. 529–590.Google Scholar
Hechler, Stephen M. [1974], On the existence of certain cofinal subsets ofωω, Axiomatic set theory (Jech, Thomas J., editor), Proceedings of symposia in pure mathematics, vol. 13, part 2, American Mathematical Society, Providence.Google Scholar
Hilbert, David [1926], Über das Unendliche, Mathematische Annalen, vol. 95, pp. 161–190, translated into French by André Weil in Acta Mathematica , vol. 48 (1926), pp. 91–122; translated in van Heijenoort [1967], pp. 367–392.Google Scholar
Jané, Ignacio [1995], The role of the absolute infinite in Cantor's conception of set, Erkenntnis, vol. 42, pp. 375–402.Google Scholar
Jech, Thomas [2002], Set theory, third millennium ed., Springer, Berlin, revised and expanded.Google Scholar
Jensen, Ronald B. [1972], The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4, pp. 229–308.Google Scholar
Kanamori, Akihiro [1995], The emergence of descriptive set theory, From Dedekind to Gödel: Essays on the development of the foundations of mathematics (Hintikka, Jaakko, editor), Synthese Library, vol. 251, Kluwer, Dordrecht, pp. 241–262.Google Scholar
Kanamori, Akihiro [1996], The mathematical development of set theory from Cantor to Cohen, this Bulletin, vol. 2, pp. 1–71.Google Scholar
Kanamori, Akihiro [2003], The higher infinite, second ed., Springer-Verlag, Heidelberg.Google Scholar
Kanamori, Akihiro [2004], Zermelo and set theory, this Bulletin, vol. 10, pp. 487–553.Google Scholar
Kennedy, Juliette C. and van Atten, Mark [2004], Gödel's modernism: On set-theoretic incompleteness, Graduate Faculty Philosophy Journal, vol. 25, pp. 289–349.Google Scholar
Koellner, Peter [2006], On the question of absolute undecidability, Philosophia Mathematica, vol. 14, no. 2.Google Scholar
Kondô, Motokiti [1939], Sur l'uniformisation des complémentaires analytiques et les ensembles projectifs de la seconde classe, Japanese Journal of Mathematics, vol. 15, pp. 197–230.Google Scholar
Krajewski, Stanisław [2004], Gödelon Tarski, Annals of Pure and Applied Logic, vol. 127, pp. 303–323.Google Scholar
Kreisel, Georg [1980], Kurt Gödel, 28 April 1906–14 January 1978, Biographical Memoirs of the Fellows of the Royal Society, vol. 26, pp. 149–224, corrections, vol. 27 (1981), p. 697 and vol. 28 (1982), p. 718.Google Scholar
Kuratowski, Kazimierz [1931], Evaluation de la classe Borélienne ou projective d'un ensemble de points à l'aide des symboles logiques, Fundamenta Mathematicae, vol. 17, pp. 249–272.CrossRefGoogle Scholar
Kuratowski, Kazimierz and Tarski, Alfred [1931], Les opérations logiques et les ensembles projectifs, Fundamenta Mathematicae, vol. 17, pp. 240–248, reprinted in Tarski [1986], vol. 1, pp. 551–559; translated in Tarski [1983], pp. 143–151.Google Scholar
Levy, Azriel [1960], Axiom schemata of strong infinity in axiomatic set theory, Pacific Journal of Mathematics, vol. 10, pp. 223–238, reprinted in Mengenlehre , Wissensschaftliche Buchgesellschaft Darmstadt, 1979, pp. 238–253.Google Scholar
Levy, Azriel and Solovay, Robert [1967], Measurable cardinals and the continuum hypothesis, Israel Journal of Mathematics, vol. 5, pp. 234–248.CrossRefGoogle Scholar
Mazurkiewicz, Stefan [1927], Sur une propriété des ensembles C(A), Fundamenta Mathematicae, vol. 10, pp. 172–174.CrossRefGoogle Scholar
Montague, Richard M. [1961], Fraenkel's addition to the axioms of Zermelo, Essays on the foundations of mathematics (Bar-Hillel, Yehoshua, E. Poznanski, I. J., Rabin, Michael O., and Robinson, Abraham, editors), Magnes Press, Jerusalem, dedicated to Professor Fraenkel, A. A. on his 70th Birthday, pp. 91–114.Google Scholar
Moschovakis, Yiannis N. [1980], Descriptive set theory, North-Holland, Amsterdam.Google Scholar
Murawski, Roman [1998], Undefinability of truth. The problem of priority: Tarskivs Gödel, History and Philosophy of Logic, vol. 19, pp. 153–160.Google Scholar
Myhill, John R. and Scott, Dana S. [1971], Ordinal definability, Proceedings of symposia in pure mathematics (Scott, Dana S., editor), Axiomatic Set Theory, vol. 13, part 1, American Mathematical Society, Providence, pp. 271–278.Google Scholar
Plotkin, Jacob M. (editor) [2005], Hausdorff on ordered sets, American Mathematical Society, Providence.Google Scholar
Ramsey, Frank P. [1925], The foundations of mathematics, Proceedings of the London Mathematical Society, vol. 25, pp. 338–384.Google Scholar
Scott, Dana S. [1955], Definitions by abstraction in axiomatic set theory, Bulletin of the American Mathematical Society, vol. 61, p. 442.Google Scholar
Scott, Dana S. [1961], Measurable cardinals and constructible sets, Bulletin de l'Academie Polonaise des Sciences, Série des Sciences Mathematiques, Astronomiques et Physiques, vol. 9, pp. 521–524.Google Scholar
Skolem, Thoralf [1920], Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen, Videnskaps-selskapets Skrifter, I, no. 4, pp. 1–36, reprinted in [1970] below, pp. 103–136. Partially translated in van Heijenoort [1967], pp. 252–263.Google Scholar
Skolem, Thoralf [1923], Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre, Matematikerkongressen i Helsingfors den 4–7 Juli 1922, Den femte skandinaviska matematikerkongressen, Redogörelse, Akademiska-Bokhandeln, Helsinki, reprinted in [1970] below, pp. 137–152; translated in van Heijenoort [1967], pp. 290–301, pp. 217–232.Google Scholar
Skolem, Thoralf [1962], Abstract set theory, Notre Dame Mathematical Lecture Notes, no. 8, Notre Dame.Google Scholar
Steel, John R. [1995], HODL(R) is a core model below Θ, this Bulletin, vol. 1, pp. 75–84.Google Scholar
Tarski, Alfred [1931], Sur les ensembles définissables de nombres réels, Fundamenta Mathematicae, vol. 17, pp. 210–239, reprinted in Tarski [1986] below, vol. 1, pp. 517–548; translated in Tarski [1983] below, pp. 110–142.Google Scholar
Tarski, Alfred [1933], Pojȩcie prawdy w jȩzykach nauk dedukcyjnych (The concept of truth in the languages of deductive sciences), Prace Towarzystwa Naukowego Warszawskiego, Wydział III, Nauk Matematyczno-fizycznych (Travaux de la Société des Sciences et des Lettres de Varsovie, Classe III, Sciences Mathematiques et Physiques), no. 34.Google Scholar
Tarski, Alfred [1935], Der Wahrheitsbegriff in den formalisierten Sprachen, German translation of [1933] with a postscript, Studia Philosophica vol. 1, pp. 261–405; reprinted in [1986] below, vol. 2, 51–198; translated in [1983] below, pp. 152–278.Google Scholar
Tarski, Alfred [1983], Logic, semantics, metamathematics. Papers from 1923 to 1938, second ed., Hackett, Indianapolis, translations by Woodger, J. H..Google Scholar
Tarski, Alfred [1986], Collected papers, (Givant, Steven R. and McKenzie, Ralph N., editors), Birkhäuser, Basel.Google Scholar
Tarski, Alfred and Vaught, Robert L. [1957], Arithmetical extensions of relational systems, Compositio Mathematica, vol. 13, pp. 81–102, reprinted in Tarski [1986], vol. 3, pp. 653–674.Google Scholar
Taub, Abraham H. (editor) [1961], John von Neumann. Collected works, vol. 1, Pergamon Press, New York.Google Scholar
van Heijenoort, Jean (editor) [1967], From Frege to Gödel. A source book in mathematical logic, 1879–1931, Harvard University Press, Cambridge.Google Scholar
von Neumann, John [1923], Zur Einführung der transfiniten Zahlen, Acta Litterarum ac Scientiarum Regiae Universitatis Hungaricae Francisco-Josephinae (Szeged), sectio scientiarum mathematicarum, vol. 1, pp. 199–208, reprinted in [1961] below, vol. 1, pp. 24–33; translated in van Heijenoort [1967], pp. 346–354.Google Scholar
von Neumann, John [1925], Eine Axiomatisierung der Mengenlehre, Journal für die reine und angewandte Mathematik, vol. 154, pp. 219–240, Berichtigung [1925], Eine Axiomatisierung der Mengenlehre, Journal für die reine und angewandte Mathematik , vol. 155, p. 128; reprinted in [1961] below, pp. 34–56; translated in van Heijenoort [1967], pp. 393–413.Google Scholar
von Neumann, John [1928], Über die Definition durch transfinite Induktion und verwandte Fragen der allgemeinen Mengenlehre, Mathematische Annalen, vol. 99, pp. 373–391, reprinted in [1961] below, vol. 1, pp. 320–338.Google Scholar
von Neumann, John [1929], Über eine Widerspruchfreiheitsfrage in der axiomatischen Mengenlehre, Journal für die reine und angewandte Mathematik, vol. 160, pp. 227–241, reprinted in [1961] below, pp. 494–508.Google Scholar
Wang, Hao [1981], Popular lectures on mathematical logic, Van Nostrand Reinhold, New York.Google Scholar
Wang, Hao [1996], A logical journey. From Gödel to Philosophy, The MIT Press, Cambridge.Google Scholar
Woleński, Jan [2005], Gödel, Tarski and truth, Revue International de Philosophie, vol. 59, no. 4, pp. 459–490.Google Scholar
Woodin, W. Hugh [1999], The axiom of determinacy, forcing axioms, and the nonstationary ideal, DeGruyter Series in Logic and Its Applications, vol. 1.Google Scholar
Zermelo, Ernst [1908], Untersuchungen über die Grundlagen der Mengenlehre I, Mathematische Annalen, vol. 65, pp. 261–281, translated in van Heijenoort [1967], pp. 199–215.Google Scholar
Zermelo, Ernst [1930], Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre, Fundamenta Mathematicae, vol. 16, pp. 29–47.Google Scholar
Zermelo, Ernst [1932], Über Stufen der Quantifikation und die Logik des Unendlichen, Jahresbericht der deutschen Mathematiker-Vereinigung (Angelegenheiten), vol. 41, pp. 85–88.Google Scholar