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Bernays and Set Theory

Published online by Cambridge University Press:  15 January 2014

Akihiro Kanamori
Akihiro Kanamori*
Affiliation:
Department of Mathematics, Boston University, 111 Cummington Street, Boston, Massachusetts 02215, USAE-mail: aki@math.bu.edu
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Abstract

We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higher-order reflection principles.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 15 , Issue 1 , March 2009 , pp. 43 - 69
Copyright
Copyright © Association for Symbolic Logic 2009

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References

REFERENCES

[1] Ackermann, Wilhelm, Begründung des “tertium non datur” mittels der Hilbertschen Theorie der Widerspuchsfreiheit, Mathematische Annalen, vol. 93 (1924), pp. 1–36.CrossRefGoogle Scholar
[2] Bernays, Paul, Über Hilberts Gedanken zur Grundlegung der Arithmetik, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 31 (1922), pp. 10–19.Google Scholar
[3] Bernays, Paul, A system of axiomatic set theory—Part I, The Journal of Symbolic Logic, vol. 2 (1937), pp. 65–77, reprinted in [57] pp. 1–13.CrossRefGoogle Scholar
[4] Bernays, Paul, A system of axiomatic set theory—Part II, The Journal of Symbolic Logic, vol. 6 (1941), pp. 1–17, reprinted in [57] pp. 14–30.CrossRefGoogle Scholar
[5] Bernays, Paul, A system of axiomatic set theory—Part III, The Journal of Symbolic Logic, vol. 7 (1942), pp. 65–89, reprinted in [57], pp. 31–55.CrossRefGoogle Scholar
[6] Bernays, Paul, A system of axiomatic set theory—Part IV, The Journal of Symbolic Logic, vol. 7 (1942), pp. 133–145, reprinted in [57] pp. 56–68.CrossRefGoogle Scholar
[7] Bernays, Paul, A system of axiomatic set theory—Part V, The Journal of Symbolic Logic, vol. 8 (1943), pp. 89–106, reprinted in [57] pp. 69–86.CrossRefGoogle Scholar
[8] Bernays, Paul, A system of axiomatic set theory—Part VI, The Journal of Symbolic Logic, vol. 13 (1948), pp. 65–79, reprinted in [57] pp. 87–101.CrossRefGoogle Scholar
[9] Bernays, Paul, A system of axiomatic set theory—Part VII, The Journal of Symbolic Logic, vol. 19 (1954), pp. 81–96, reprinted in [57], pp. 102–117.CrossRefGoogle Scholar
[10] Bernays, Paul, Axiomatic Set Theory, Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1958, with a historical introduction by Fraenkel, Abraham A..Google Scholar
[11] Bernays, Paul, Die hohen Unendlichkeiten und die Axiomatik der Mengenlehre, Infinitistic methods. Proceedings of the symposium on foundations of mathematics, Pergamon Press, Oxford, 1961, pp. 11–20.Google Scholar
[12] Bernays, Paul, Zur Frage der Unendlichkeitsschemata in der axiomatischen Mengenlehre, Essays on the foundations of mathematics (Bar-Hillel, , Poznanski, , Rabin, , and Robinson, , editors), Magnes Press, Jerusalem, 1961, dedicated to Professor Abraham A. Fraenkel on his 70th birthday, pp. 3–49.Google Scholar
[13] Bernays, Paul, What do some recent results in set theory suggest?, Problems in the philosophy of mathematics (Lakatos, Imre, editor), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1967, pp. 109–112.CrossRefGoogle Scholar
[14] Bernays, Paul, Paul Bernays—a short biography, Sets and classes (Müller, Gert H., editor), Studies in Logic and the Foundations of Mathematics, vol. 84, North-Holland, Amsterdam, 1976, This is a translation from the German, on pages xi–xiii., pp. xiv–xvi.Google Scholar
[15] Burgess, John P., Fixing Frege, Princeton University Press, Princeton, 2005.CrossRefGoogle Scholar
[16] Dawson, John W. Jr., Logical dilemmas: The life and work of Kurt Gödel, A.K. Peters, Wellesley, 1997.Google Scholar
[17] Ebbinghaus, Heinz-Dieter, Ernst Zermelo. An approach to his life and work, Springer, Berlin, 2007.Google Scholar
[18] Ewald, William, Sieg, Wilfried, and Majer, Ulrich (editors), Hilbert's lectures on the foundations of arithmetic and logic, 1917-1933, Springer, 2009.Google Scholar
[19] Felgner, Ulrich, Ein Brief Gödels zum Fundierungsaxiom, Kurt Gödel—Wahrheit & Beweisbarkeit. Kompendium zum Werk (Buldt, Bernd et al., editors), öbv et hpt, Wien, 2002, pp. 205–213.Google Scholar
[20] Fraenkel, Abraham A., Über die Zermelosche Begründung der Mengenlehre, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 30II (1921), pp. 97–98, abstract.Google Scholar
[21] Fraenkel, Abraham A., Zu den Grundlagen der Cantor–Zermeloschen Mengenlehre, Mathematische Annalen, vol. 86 (1922), pp. 230–237.CrossRefGoogle Scholar
[22] Gödel, Kurt, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter System I, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173–198, reprinted and translated in [26] pp. 144–195.CrossRefGoogle Scholar
[23] Gödel, Kurt, Über Vollständigkeit und Widerspruchsfreiheit, Ergebnisse eines mathematischen Kolloquiums, vol. 3 (1932), pp. 12–13, reprinted and translated in [26], pp. 234–237.Google Scholar
[24] Gödel, Kurt, Consistency-proof for the generalized continuum-hypothesis, Proceedings of the National Academy of Sciences USA, (1939), pp. 220–224, reprinted in [27] pp. 28–32.Google Scholar
[25] Gödel, Kurt, 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, 1940, reprinted in [27] pp. 33–101.Google Scholar
[26] Gödel, Kurt, Collected works, volume I: Publications 1929–1936, Clarendon Press, Oxford, 1986, edited by Feferman, Solomon et al. Google Scholar
[27] Gödel, Kurt, Collected works, volume II: Publications 1938–1974, Oxford University Press, New York, 1990, edited by Feferman, Solomon et al. Google Scholar
[28] Gödel, Kurt, Collected works, volume III: Unpublished essays and lectures, Oxford University Press, New York, 1995, edited by Feferman, Solomon et al. Google Scholar
[29] Gödel, Kurt, Collected works, volume IV: Correspondence A–G, Clarendon Press, Oxford, 2003, edited by Feferman, Solomon et al. Google Scholar
[30] Gödel, Kurt, Collected works, volume V: Correspondence H–Z, Clarendon Press, Oxford, 2003, edited by Feferman, Solomon et al. Google Scholar
[31] Hallett, Michael, Cantorian set theory and limitation of size, Logic Guides, no. 10, Clarendon Press, Oxford, 1984.Google Scholar
[32] Hanf, William P., Incompactness in languages with infinitely long expressions, Fundamenta Mathematicae, vol. 53 (1964), pp. 309–324.CrossRefGoogle Scholar
[33] Hanf, William P. and Scott, Dana S., Classifying inaccessible cardinals, Notices of the American Mathematical Society, vol. 8 (1961), p. 445, abstract.Google Scholar
[34] Hauschild, Kurt, Bemerkungen, das Fundierungsaxiom betreffend, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 51–56.CrossRefGoogle Scholar
[35] Hilbert, David, Über die Grundlagen der Logik und der Arithmetik, Verhandlungen des dritten internationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 13. August 1904 (Krazer, Adolf, editor), Teubner, Leipzig, 1905, translated in [70], pages 130–138, pp. 129–138.Google Scholar
[36] Hilbert, David, Axiomatisches Denken, Mathematische Annalen, vol. 78 (1918), pp. 405–415.CrossRefGoogle Scholar
[37] Hilbert, David, Neubegründung der Mathematik (erste Mitteilung), Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität, vol. 1 (1922), pp. 157–177.CrossRefGoogle Scholar
[38] Hilbert, David, Über das Unendliche, Mathematische Annalen, (1926), pp. 161–190, translated in [70] pp. 367–392.Google Scholar
[39] Hilbert, David and Bernays, Paul, Grundlagen der Mathematik I, Springer-Verlag, Berlin, 1934, second edition, 1968.Google Scholar
[40] Hilbert, David, Grundlagen der Mathematik II, Springer-Verlag, Berlin, 1939, second edition, 1970.Google Scholar
[41] Kanamori, Akihiro, Zermelo and set theory, this Bulletin, vol. 10 (2004), pp. 487–553.Google Scholar
[42] Kanamori, Akihiro, Gödel and set theory, this Bulletin, vol. 13 (2007), pp. 153–188.Google Scholar
[43] Kuratowski, Kazimierz, Sur la notion de l'ordre dans la théorie des ensembles, Fundamenta Mathematicae, vol. 2 (1921), pp. 161–171.CrossRefGoogle Scholar
[44] Levy, Azriel, Axiom schemata of strong infinity in axiomatic set theory, Pacific Journal of Mathematics, vol. 10 (1960), pp. 223–238.CrossRefGoogle Scholar
[45] Levy, Azriel, On von Neumann's axiom system for set theory, American Mathematical Monthly, (1968), pp. 762–763.Google Scholar
[46] Mahlo, Paul, Über lineare transfinite Mengen, Berichte über die Verhandlungen der Königlich-Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse, vol. 63 (1911), pp. 187–225.Google Scholar
[47] Levy, Azriel, Zur Theorie und Anwendung der ρ0-Zahlen, Berichte über die Verhandlungen der Königlich-Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse, vol. 64 (1912), pp. 108–112.Google Scholar
[48] Levy, Azriel, Zur Theorie und Anwendung der ρ0-Zahlen II, Berichte über die Verhandlungen der Königlich-Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse, vol. 65 (1913), pp. 268–282.Google Scholar
[49] Mancosu, Paolo, From Brouwer to Hilbert, Oxford University Press, New York, 1998.Google Scholar
[50] Meschkowski, Herbert, Probleme des Unendlichen. Werk und Leben Georg Cantors, Friedr. Vieweg & Sohn, Braunschweig, 1967.CrossRefGoogle Scholar
[51] Mirimanoff, Dmitry, Les antinomies de Russell et de Burali-Forti et le problème fondamental de la théorie des ensembles, L'Enseignment Mathématique, vol. 19 (1917), pp. 37–52.Google Scholar
[52] Mirimanoff, Dmitry, Remarques sur la théorie des ensembles et les antinomies Cantoriennes. I, L'Enseignment Mathématique, vol. 19 (1917), pp. 209–217.Google Scholar
[53] Mirimanoff, Dmitry, Remarques sur la théorie des ensembles et les antinomies Cantoriennes. II, L'Enseignment Mathématique, vol. 21 (1919), pp. 29–52.Google Scholar
[54] Montague, Richard M., Fraenkel's addition to the axioms of Zermelo, Essays on the foundations of mathematics (Bar-Hillel, , Poznanski, , Rabin, , and Robinson, , editors), Magnus Press, Jerusalem, 1961, dedicated to Professor A. A. Fraenkel on his 70th birthday, pp. 91–144.Google Scholar
[55] Moore, Gregory H., Bernays, Paul Isaac, Dictionary of scientific biography, vol. 17, Supplement II, Charles Scribner's Sons, New York, 1981, pp. 75–78.Google Scholar
[56] Mostowski, Andrzej, Some impredicative definitions in the axiomatic set theory, Fundamenta Mathematicae, vol. 37 (1951), pp. 111–124.CrossRefGoogle Scholar
[57] Müller, Gert H. (editor), Sets and classes. on the work of Paul Bernays, Studies in Logic and the Foundations of Mathematics, vol. 84, North-Holland, Amsterdam, 1976.Google Scholar
[58] Novak, Ilse, A construction for models of consistent systems, Fundamenta Mathematicae, vol. 37 (1951), pp. 87–110.CrossRefGoogle Scholar
[59] Parsons, Charles, Paul Bernays' later philosophy of mathematics, Logic Colloquium '05 (Cambridge) (Dimitracopoulos, Costas et al., editors), Lecture Notes in Logic, vol. 28, Cambridge University Press, 2008, pp. 129–150.Google Scholar
[60] Robinson, Abraham, Recent developments in model theory, Logic, methodology and philosophy of science. Proceedings of the 1960 international congress, Stanford (Nagel, Ernest, Suppes, Patrick, and Tarski, Alfred, editors), Stanford University Press, Stanford, 1962.Google Scholar
[61] Robinson, Raphael M., The theory of classes, amodification of von Neumann's system, The Journal of Symbolic Logic, vol. 2 (1937), pp. 29–36.CrossRefGoogle Scholar
[62] Shoenfield, Joseph R., A relative consistency proof, The Journal of Symbolic Logic, vol. 19 (1954), pp. 21–28.CrossRefGoogle Scholar
[63] Sieg, Wilfried, Hilbert's programs: 1917–1922, this Bulletin, vol. 5 (1999), pp. 1–44.Google Scholar
[64] Sieg, Wilfried, Only two letters: The correspondence between Herbrand and Gödel, this Bulletin, vol. 11 (2005), pp. 172–184.Google Scholar
[65] Sieg, Wilfried and Tait, William W. (editors), Paul Bernays: Essays in the philosophy of mathematics, Open Court, Chicago, 2009.Google Scholar
[66] Skolem, Thoralf, Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre, Matematikerkongressen i Helsingfors den 4–7 Juli 1922, Den femte skandinaviska matematikerkongressen, Redogörelse, Akademiska-Bokhandeln, Helsinki, 1923, translated in [70], pages 290–301, pp. 217–232.Google Scholar
[67] Specker, Ernst, Zur Axiomatik der Mengenlehre (Fundierungs- und Auswahlaxiom), Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 3 (1957), pp. 173–210.CrossRefGoogle Scholar
[68] Teichmüller, Oswald, Braucht der Algebraiker das Auswahlaxiom?, Deutsche Mathematik, vol. 4 (1939), pp. 567–577, reprinted in [69] pp. 323–333.Google Scholar
[69] Teichmüller, Oswald, Gesammelte Abhandlungen. Collected papers, (Ahlfors, Lars V. and Gehring, Frederick W., editors), Springer-Verlag, Berlin, 1982.Google Scholar
[70] van Heijenoort, Jean (editor), From Frege to Gödel. A source book in mathematical logic, 1879–1931, Harvard University Press, Cambridge, 1967.Google Scholar
[71] von Neumann, John, Zur Einführung der transfiniten Zahlen, Acta Litterarum ac Scientiarum Regiae Universitatis Hungaricae Fancisco-Josephinae, sectio scientiarum mathematicarum, vol. 1 (1923), pp. 199–208, translated in [70], pages 346–354.Google Scholar
[72] von Neumann, John, Eine Axiomatisierung der Mengenlehre, Journal für die reine und angewandte Mathematik, vol. 154 (1925), pp. 219–240, translated in [70], pages 393–413.Google Scholar
[73] von Neumann, John, Zur Hilbertschen Beweistheorie, Mathematische Zeitschrift, vol. 26 (1927), pp. 1–46.CrossRefGoogle Scholar
[74] von Neumann, John, Über eine Widerspruchfreiheitsfrage in der axiomatischen Mengenlehre, Journal für die reine und angewandte Mathematik, vol. 160 (1929), pp. 227–241.CrossRefGoogle Scholar
[75] Vopěnka, Petr and Hajek, Petr, Über die Gültigkeit des Fundierungsaxioms in speziellen Systemen der Mengentheorie, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 9 (1963), pp. 235–241.CrossRefGoogle Scholar
[76] Zach, Richard, Completeness before Post: Bernays, Hilbert, and the development of propositional logic, this Bulletin, vol. 5 (1999), pp. 331–366.Google Scholar
[77] Zermelo, Ernst, Beweis, daß jede Menge wohlgeordnet werden kann. (Aus einem an Herrn Hilbert gerichteten Briefe), Mathematische Annalen, vol. 59 (1904), pp. 514–516, translated in [70], pages 139–141.CrossRefGoogle Scholar
[78] Zach, Richard, Untersuchungen über die Grundlagen der Mengenlehre I, Mathematische Annalen, vol. 65 (1908), pp. 261–281, translated in [70], pages 199–215.Google Scholar
[79] Zach, Richard, Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre, Fundamenta Mathematicae, vol. 16 (1930), pp. 29–47.Google Scholar