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The Burali-Forti Paradox

Published online by Cambridge University Press:  14 March 2022

Irving M. Copi
Irving M. Copi*
Affiliation:
University of Michigan
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Extract

The year 1897 saw the publication of the first of the modern logical paradoxes. It was published by Cesare Burali-Forti, the Italian mathematician whose name it has come to bear. Burali-Forti's own formulation of the paradox was not altogether satisfactory, as he had confused well-ordered sets as defined by Cantor with what he himself called “perfectly ordered sets” (“Classe parfettamente ordinata”). However, he soon realized his mistake, and published a note admitting the error and making the correction. He concluded the note with the observation that his result could be established on the basis of the correct definition of well-ordered set as easily as for the “perfectly ordered sets” for which it had first been obtained. We shall reproduce his results in their corrected form.

Type
Research Article
Information
Philosophy of Science , Volume 25 , Issue 4 , October 1958 , pp. 281 - 286
Copyright
Copyright © 1958, The Williams & Wilkins Company

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Footnotes

1

From a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan.

References

1. Bell, E. T. The Development of Mathematics, New York and London, 2nd ed., 1945.Google Scholar
2. Berstein, Felix, “Uber die Reihe der Transfiniten Ordnungszahlen”, Mathematische Annalen, 60 (1905), 187–193.CrossRefGoogle Scholar
3. Black, Max, The Nature of Mathematics, London, 1933, New York, 1934.Google Scholar
4. Burali-Forti, Cesare, “Una Questione sui Numeri Transfiniti”, Rendiconti del Circolo Matematico di Palermo, 11 (1897), 154–164.CrossRefGoogle Scholar
5. Burali-Forti, Cesare, “Sulle Classi ben Ordinate”, Ibid., 260.CrossRefGoogle Scholar
6. Georg Cantor Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, ed. by Ernst Zermelo. Berlin, 1932.CrossRefGoogle Scholar
7. Church, Alonzo, A Bibliography of Symbolic Logic, Providence, 1938. Reprinted from The Journal of Symbolic Logic, 1 (1936). vol. 3 (1938).Google Scholar
8. Church, Alonzo, “Logical Paradoxes”, The Dictionary of Philosophy, ed. by D. Runes, New York, 1942, 224–225.Google Scholar
9. Couturat, Louis, “Pour la logistique”, Revue de Metaphysique et de Morale, 14 (1906), 208–250.Google Scholar
10. Fraenkel, Adolf, Einleitung in die Mengenlehre, 3rd ed., Berlin, 1928.CrossRefGoogle Scholar
11. Fraenkel, Adolf, “Das Leben Georg Cantors”, Georg Cantor Gesammelte Abhandlungen, (6), 452–483.Google Scholar
12. Adolf Fraenkel, Review of Jean Cavailles' “Remarques sur la Formation de la Theorie Abstraite des Ensembles”, Journal of Symbolic Logic, 3 (1938), 167–168.CrossRefGoogle Scholar
13. Grelling, Kurt, Mengenlehre, Leipzig and Berlin, 1924.Google Scholar
14. Grelling, Kurt and Nelson, Leonard, “Bemerkungen zu den Paradoxien von Russell und Burali-Forti”, Abhandlung der Fries'sche Schule, n.s. 2 (1907-8), 300–324.Google Scholar
15. Hagström, K. G., “Note sur l'antinomie Burali-Forti.” Arkiv for Matematik, Astronomi och Fysik, 10 (1914), 1–4.Google Scholar
16. Hessenberg, Gerhard, “Grundbegriffe der Mengenlehre”, Abhandlung der Fries'schen Schule, n.s. vol. 1 (1906). Reprinted Gottingen, 1906.Google Scholar
17. Hilbert, David, “Uber den Zahlbegriff”, Jahresbericht der Deutschen Mathematiker-Vereinigung, 8 (1900), 180–184.Google Scholar
18. Hobson, E. W., “On the General Theory of Transfinite Numbers and Order Types”, Proceedings of the London Mathematical Society, 2nd series, 3 (1905), 170–188.Google Scholar
19. Hobson, E. W., The Theory of Functions of a Real Variable and the Theory of Fourier's Series. 2nd ed., vol. 1, Cambridge, 1921.Google Scholar
20. Jörgensen, Jörgen, A Treatise of Formal Logic. Copenhagen and London, 1931, 3 vols.Google Scholar
21. Jourdain, P. E. B., “On the Transfinite Cardinal Numbers of Well-Ordered Aggregates”, Philosophical magazine, 6 s., 7 (1904), 61–75.Google Scholar
22. Jourdain, P. E. B. “On a Proof that Every Aggregate Can Be Well-Ordered,” Mathematische Annalen, 60 (1905), 465–470.CrossRefGoogle Scholar
23. Jourdain, P. E. B., “On the Question of the Existence of Transfinite Numbers”, Proceedings of the London Mathematical Society, 2 s., 4 (1906), 266–283.Google Scholar
24. Kleene, Stephen Cole, Introduction to Metamathematics, New York and Toronto, 1952.Google Scholar
25. König, Julius, Neue grundlagen der Logik, Arithmetik und Mengenlehre, Leipzig, 1914.CrossRefGoogle Scholar
26. Mirimanoff, Dmitry, “Les Antinomies de Russell et de Burali-Forti et le Problems Fondamental de la Theorie des Ensembles”, L'Enseignement mathematique, 19 (1917), 37–52.Google Scholar
27. Olivier, Louis, “La Theorie des Ensembles”, Revue générale des sciences, 16 (1905), 241–242.Google Scholar
28. Poincaré, Henri, “Les Mathématiques et la Logique”, Revue de Metaphysique et de Morale, 13 (1905), 815-835; 14 (1906), 17-34, 294–317.Google Scholar
29. Poincaré, Henri, Science et Methode. Translated as part 3 of The Foundations of Science, Lancaster, Pa., 1946.Google Scholar
30. Rosser, J. Barkley, “The Burali-Forti Paradox”, The Journal of Symbolic Logic, 7 (1942), 1–17.CrossRefGoogle Scholar
31. Russell, Bertrand, “Theorie Generale des Series Bien-ordonnes”, Revista de Mathematica, 8 (1902-1906), 12–43.Google Scholar
32. Russell, Bertrand, Principles of Mathematics, Cambridge, 1903. 2nd ed., New York, 1938.Google Scholar
33. Russell, Bertrand, “On Some Difficulties in the Theory of Transfinite Numbers and Order Types”, Proceedings of the London Mathematical Society, 2 s., 4 (1906), 29–53.Google Scholar
34. Russell, Bertrand, “Mathematical Logic as Based on the Theory of Types”, American Journal of Mathematics, 30 (1908), 222-262. Reprinted in Logic and Knowledge, ed. by R. C. Marsh, London, 1956, pp. 57–102.CrossRefGoogle Scholar
35. Russell, Bertrand, The Philosophy of Mr. B*rtr*nd R*ss*ll. Chicago, 1918.Google Scholar
36. Rustow, A., Der Lügner, Leipzig, 1910.Google Scholar
37. Whitehead, A. N. and Russell, B., Principia Mathematica, Cambridge, vol. 1, 1910; vol. 2, 1912; vol. 3, 1913. 2nd ed., 1925-1927.Google Scholar
38. Young, W. H., “Presidential Address”, Proceedings of the London Mathematical Society, 2 s., 24 (1925), 421–434.Google Scholar
39. Young, W. H. and Young, G. C., Review of Hobson's Theory of Functions of a Real Variable. Mathematical Gazette, 14 (1928), 98-104Google Scholar