Skip to main content

Structures

  • Chapter
Book cover Theory of Sets
  • 2827 Accesses

Abstract

The purpose of this chapter is to describe once and for all a certain number of formative constructions and proofs (cf. Chapter I, § 1, no. 3 and § 2, no. 2) which arise very frequently in mathematics.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. O. Neugebauer, Vorlesungen über die Geschichte der antiken Mathematik, Vol. I, Vorgriechische Mathematik, Berlin (Springer), 1934.

    Google Scholar 

  2. The Works of Aristotle, translated under the editorship of J. A. Smith and W. D. Ross (12 volumes, Oxford, 1908–1952).

    Google Scholar 

  3. (bis). T. L. Heath, Mathematics in Aristotle, Oxford (Clarendon Press), 1949.

    MATH  Google Scholar 

  4. T. L. Heath, The Thirteen Books of Euclid’s Elements…, 3 volumes, Cambridge, 1908.

    Google Scholar 

  5. Archimedes Opera Omnia, 3 volumes, edited by J. L. Heiberg, 2nd edition, 1913–1915.

    Google Scholar 

  6. (bis). T. L. Heath, The Method of Archimedes, Cambridge, 1912.

    Google Scholar 

  7. J. M. Bochenski, Ancient Formal Logic, Studies in Logic, Amsterdam (North Holland Pub. Co.), 1951.

    MATH  Google Scholar 

  8. P. Böhner, Medieval Logic, an Outline of its Development from 1250to ca. 1400, Chicago, 1952.

    Google Scholar 

  9. D. Francisci Maurolyci, Abbatis Messamensis, Mathematici Celeberrimi, Arithmeticorum Libri Duo, Venice, 1575.

    Google Scholar 

  10. Galileo Galilei, Opere, Ristampa della Edizione Nazionale, 20 volumes, Florence (Barbara), 1929–39.

    Google Scholar 

  11. A. Girard, Invention nouvelle en l’Algèbre, 1629 (new edition, Bierens de Haan, 1884).

    Google Scholar 

  12. R. Descartes, Ceuvres, edited by C. Adam and P. Tannery, 11 volumes, Paris (L. Cerf), 1897–1909.

    Google Scholar 

  13. B. Pascal, Euvres, edited by Brunschvicg, 14 volumes, Paris (Hachette), 1904–1914.

    Google Scholar 

  14. G. W. Leibntz: Mathematische Schriften, edited by C. I. Gerhardt, 7 volumes, Berlin-Halle (Asher-Schmidt), 1849–1863;

    Google Scholar 

  15. G. W. Leibntz: Philosophische Schriften, edited by C. I. Gerhardt, 7 volumes, Berlin, 1840–1890

    Google Scholar 

  16. G. W. Leibntz: Opuscules et fragments inédits, edited by L. Couturat, Paris (Alcan), 1903.

    Google Scholar 

  17. (bis). L. ?outurat, La logique de Leibniz d’après des documents inédits, Paris (Alcan), 1901.

    MATH  Google Scholar 

  18. D’Alembert, Encyclopédie, Paris, 1751–1765, articles “Négatif”, “Imaginaire”, “Définition”.

    Google Scholar 

  19. C. F. Gauss, Werke, 12 volumes, Göttingen, 1870–1927.

    Google Scholar 

  20. N. Lobatschevsky, Pangeometrie, Ostwald’s Klassiker, no. 130, Leipzig (Engelmann), 1902.

    Google Scholar 

  21. G. Boole: The Mathematical Analysis of Logic, Cambridge-London, 1847 (= Collected Logical Works, edited by P. Jourdain, Chicago-London, 1916, vol. I)

    Google Scholar 

  22. G. Boole: An Investigation of the Laws of Thought, Cambridge-London, 1854 (= Collected Logical Works, vol. II).

    Google Scholar 

  23. H. Gkjissmann, Gesammelte Werke, 6 volumes, Leipzig (Teubner), 1894.

    Google Scholar 

  24. B. Bolzano, Paradoxien des Unendlichen, Leipzig, 1851.

    Google Scholar 

  25. B. Riemann, Gesammelte mathematische Werke, 2nd edition, Leipzig (Teubner), 1892.

    MATH  Google Scholar 

  26. H. Hankel, Theorie der complexen,Zahlensysteme, Leipzig (Voss), 1867.

    Google Scholar 

  27. A. De Morgan:“On the syllogism (III)”, Trans. Garb. Phil. Soc., 10 (1858), pp. 173–230

    Google Scholar 

  28. A. De Morgan “On the syllogism (IV) and on the logic of relations”, Trans. Garb. Phil. Soc., 10 (1860) pp. 331–358.

    Google Scholar 

  29. C. S. Peirce “Upon the logic of mathematics”, Proc. Amer. Acad. Arts and Sci., 7 (1865–1868), pp. 402–412

    Article  Google Scholar 

  30. C. S. Peirce “On the algebra of logic”, Amer. Journ. of Math., 3 (1880), pp. 49–57

    Article  MathSciNet  Google Scholar 

  31. C. S. Peirce“On the algebra of logic”, Amer. Journ. of Math., 7 (1884), pp. 190–202.

    Google Scholar 

  32. E. Schröder, Vorlesungen über die Algebra der Logik, 3 volumes, Leipzig (Teubner), 1890.

    Google Scholar 

  33. G. Frege, Begriffsschrift eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle, 1879;

    Google Scholar 

  34. G. Frege Die Grundlagen der Arithmetik, 2nd edition with an English translation by J. L. Austin, New York, 1950

    Google Scholar 

  35. G. Frege Grundgesetze der Arithmetik, begrif ffsschriftlich abgeleitet, 2 volumes, Jena, 1893–1903.

    Google Scholar 

  36. G. Cantor, Gesammelte Abhandlungen, Berlin (Springer), 1932.

    Google Scholar 

  37. (bis). G. Cantor, R. Dedekind, Briefwechsel, edited by J. Cavaillès and E. Noether, Act. Sci. et Ind., no. 518, Paris (Hermann), 1937.

    Google Scholar 

  38. R. Dedekind, Gesammelte mathematische Werke, 3 volumes, Braunschweig (Vieweg), 1932.

    Google Scholar 

  39. C. Hermrte, T. Stieltjes, Correspondance, 2 volumes, Paris (Gauthier-Villars), 1905.

    Google Scholar 

  40. M. Pasch und M. Dehn, Vorlesungen über neuere Geometrie, 2nd edition, Berlin, (Springer), 1926.

    MATH  Google Scholar 

  41. G. Peauo, Arithmeticae principia, novo modo exposita, Turin, 1889

    Google Scholar 

  42. G. Peauo, I principii di Geometria, logicamente espositi, Turin, 1889

    Google Scholar 

  43. G. Peauo, Formulaire de Mathematiques, 5 volumes, Turin, 1895–1905.

    Google Scholar 

  44. D. Hilbert, Grundlagen der Geometrie, 7th edition, Leipzig-Berlin (Teubner), 1930.

    MATH  Google Scholar 

  45. D. Hilbert, Gesammelte Abhandlungen, vol. III, Berlin (Springer), 1935.

    Google Scholar 

  46. D. Hilbert, W. Ackermann, Grundzüge der theoretischen Logik, 3rd edition, Berlin (Springer), 1949.

    MATH  Google Scholar 

  47. H. Poincaré: Science et hypothése, Paris (Flammarion), 1906

    Google Scholar 

  48. E. Poincaré Science et méthode, Paris (Flammarion), 1920.

    Google Scholar 

  49. J. Richard, “Les principes des mathématiques et le problète des ensembles”, Rev. Gen. des Sci. Pures et Appl, 16 (1905), pp. 541–543.

    Google Scholar 

  50. R. Baire, E. Borel, J. Hadamard, H. Lebesgue, “Cinq lettres sur la théorie des ensembles”, Bull. Soc. Math. de France, 33 (1905), pp. 261–273.

    MATH  Google Scholar 

  51. E. Zermelo, “Beweis dass jede Menge wohlgeordnet werden kann”, Math. Ann., 59 (1904), pp. 514–516

    Article  MathSciNet  MATH  Google Scholar 

  52. E. Zermelo“Neuer Beweis fur die Möglichkeit einer Wohlordnung”, Math. Ann, 65 (1908), pp. 107–128

    Article  MATH  Google Scholar 

  53. E. Zermelo“Untersuchungen über die Grundlagen der Mengenlehre”, Math. Ann, 65 (1908), pp. 261–281.

    Article  MathSciNet  MATH  Google Scholar 

  54. B. Russell and A. N. Whitehead, Principia Mathematica, 3 volumes, Cambridge, 1910–1913.

    Google Scholar 

  55. L. E. J. Brouwer, “Intuitionism and formalism”, Bull. Amer. Math. Soc., 20 (1913), pp. 81–96

    Article  MathSciNet  MATH  Google Scholar 

  56. L. E. J. Brouwer“Zur Begründung des intuitionistischen Mathematik”, Math. Ann., 93 (1925), pp. 244–257; 95 (1926), pp. 453–473; 96 (1926), pp. 451–458.

    Google Scholar 

  57. T. Skolem, “Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre”, Wiss. Vorträge, 5 Kongress Skand. Math., Helsingfors, 1922.

    Google Scholar 

  58. K. Kuratowski, “Une méthode d’élimination des nombres transfinis des raisonnements mathématiques”, Fund. Math, 5 (1922), pp. 76–108.

    Google Scholar 

  59. A. Fraenkel “Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre”, Math. Ann., 86 (1922), pp. 230–237

    Article  MathSciNet  MATH  Google Scholar 

  60. A. Fraenkel Zehn Vorlesungen über die Grundlgaung der Mengenlehre, Wiss. und Hypothese vol. 31, Leipzig-Berlin, 1927

    Google Scholar 

  61. A. Fraenkel Einleitung in die Mengenlehre, 3rd edition, Berlin (Springer), 1928.

    MATH  Google Scholar 

  62. J. Herbrand, “Recherches sur la théorie de la démonstration”, Tray. Soc. Sci. et Lett. Varsovie, cl. II (1930), pp. 33–160.

    Google Scholar 

  63. J. Von Neumann “Eine Axiomatisierung der Mengenlehre”, Crepe, 154 (1925), pp. 219–240

    Google Scholar 

  64. J. Von Neumann “Die Axiomatisierung der Mengenlehre”, Math.,Zeitsсhr., 27 (1928), pp. 669–752

    Article  MATH  Google Scholar 

  65. J. Von Neumann“Zur Hilbertschen Beweistheorie”, Math.Zeitschr., 26 (1927), pp. 1–46.

    Article  MATH  Google Scholar 

  66. K. Godel: “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme”, Monatsh. für Math. u. Phys., 38 (1931), pp. 173–198

    Article  MathSciNet  Google Scholar 

  67. K. Godel The consistency of the axiom of choice and of the generalized continuum hypothesis, Ann. of Math. Studies, no. 3, Princeton, 1940.

    Google Scholar 

  68. M. Zorn, “A remark on method in transfinite algebra”, Bull. Amer. Math. Soc., 41 (1935), pp. 667–670.

    Article  MathSciNet  Google Scholar 

  69. A. Heyting, Mathematische Grundlagenforschung. Intuitionismus. Beweistheorie, Ergebnisse der Math., vol. 3, Berlin (Springer), 1934.

    Google Scholar 

  70. G. Gentzen, Die gegenwartige Lage in der mathematischen Grundlagenforschung. Neue Fassung des Widerspruchfreiheitsbвweises fйr die reine Zahlentheorie, Forschungen der Logik…, Нefc 4, Leipzig (Hirzel), 1938.

    Google Scholar 

  71. S. Kleene, Introduction to Metamathematics, New York, 1952.

    MATH  Google Scholar 

  72. P. J. Cohen, “The independence of the continuum hypothesis”, Proc. Nat. Acad. Sci., 50 (1963), pp. 1143–1148 and 51 (1964), pp. 105–110.

    Google Scholar 

Download references

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Bourbaki, N. (2004). Structures. In: Theory of Sets. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59309-3_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-59309-3_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-22525-6

  • Online ISBN: 978-3-642-59309-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics