The Notion of Set

  • Johan Georg Granström
Part of the Logic, Epistemology, and the Unity of Science book series (LEUS, volume 22)


The notion of set is central to modern foundations of mathematics , regardless of school. In fact, the position taken on this notion highlights major differences between the schools, but remains central to all of them. The history of the definition of this notion is the history of how universals made into objects of thought are brought into the language of logic proper, ie, brought from the metalanguage to the object language .


Inference Rule Type Theory Equality Rule Decimal Number Conceivable Object 
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  1. Troelstra, A. S. Principles of Intuitionism. Springer, 1969.Google Scholar
  2. De Morgan, A. ‘On the Structure of the Syllogism’. In: Trans. Cambridge Philos. Soc. 8 (1846), pp. 379–408.Google Scholar
  3. Symbolic Logic. London-New York: Macmillan & Co., 1896.Google Scholar
  4. Analytica posteriora. Trans. by H. Tredennick. Loeb classical library 391. Harvard University Press, 1960, pp. 1–261.Google Scholar
  5. Brown, H. C. ‘The Logic of Mr. Russell’. In: J. Philos. Psychol. Sci. Meth. 8.4 (1911), pp. 85–91.Google Scholar
  6. — ‘Beiträge zur Begründung der transfiniten Mengenlehre’. In: Math. Ann. 46.4 (1895), pp. 481–512.Google Scholar
  7. Intuitionistic Type Theory. Studies in Proof Theory. Napoli: Bibliopolis, 1984.Google Scholar
  8. — ‘Constructive mathematics and computer programming’. In: Logic, Methodology and Philosophy of Science VI. Ed. by L. J. Cohen et al. Amsterdam: North-Holland, 1982, pp. 153–175.Google Scholar
  9. Cardelli, L. and P. Wegner. ‘On Understanding Types, Data Abstraction, and Polymorphism’. In: ACM Computing Surveys 17.4 (1985), pp. 471–522.CrossRefGoogle Scholar
  10. Intuitionism : An Introduction. Amsterdam: North-Holland, 1956.Google Scholar
  11. Paradoxes of the Infinite. Trans. by D. A. Steele. London: Routledge, 1950.Google Scholar
  12. — ‘An intuitionistic theory of types : predicative part’. In: Logic Colloquium ’73. Ed. by H. E. Rose and J. Shepherdson. Amsterdam: North-Holland, 1975, pp. 73–118.Google Scholar
  13. Bishop, E. and D. Bridges. Constructive Analysis. Springer, 1985.Google Scholar
  14. Hofmann, M. ‘Extensional concepts in intensional type theory’. PhD thesis. LFCS Edinburgh, 1995.Google Scholar
  15. Boole, G. ‘The Calculus of Logic’. In: Cambridge and Dublin Math. J. 3 (1848), pp. 183–198.Google Scholar
  16. Cantor, G. ‘Über unendliche, lineare Punktmannichfaltigkeiten’. In: Math.Ann. 20.1 (1882), pp. 113–121.CrossRefGoogle Scholar
  17. — ‘The Nature and Meaning of Numbers’. In: Essays on the Theory of Numbers. Trans. by W. W. Beman. New York: Dover, 1901.Google Scholar
  18. — ‘Untersuchungen über die Grundlagen der Mengenlehre—I’. In: Math. Ann. 65 (1908), pp. 261–281.Google Scholar
  19. Peano, G. Arithmetices Principia Nova Methodo Exposita. Turin: Fratelli Bocca, 1889.Google Scholar
  20. Whitehead, A. N. and B. Russell. Principia Mathematica. Vol. 1. Cambridge University Press, 1910.Google Scholar
  21. — ‘A survey of the project Automath’. In: To H. B. Curry : Essays in CombinatoryLogic, Lambda Calculus and Formalism. Academic Press, 1980, pp. 589–606.Google Scholar
  22. Peirce, C. S. ‘On the Algebra of Logic’. In: Amer. J. Math. 3.1 (1880), pp. 15–57.CrossRefGoogle Scholar
  23. Metaphysics. Trans. by H. Tredennick. Loeb Classical Library 271 & 287. Harvard University Press, 1933 & 1935.Google Scholar
  24. In duodecim libros Metaphysicorum Aristotelis expositio. Ed. by M.-R. Cathala and R. M. Spiazzi. Turin: Marietti, 1950.Google Scholar
  25. Frege, G. Begriffsschrift. Halle: Louis Nebert, 1879.Google Scholar
  26. — ‘On Concept and object’. In: Translations from the Philosophical Writings of Gottlob Frege. Trans. by P. T. Geach. Oxford: B. Blackwell, 1960, pp. 42– 55.Google Scholar
  27. — ‘Mathematical Logic as Based on the Theory of Types’. In: Amer. J. Math. 30.3 (1908), pp. 222–262.Google Scholar
  28. Brouwer, L. E. J. ‘Intuitionism and formalism’. Trans. by A. Dresden. In: Bull.Amer. Math. Soc. 20.2 (1913), pp. 81–96.CrossRefGoogle Scholar
  29. Poinsot, J. The Material Logic of John of St. Thomas. Basic Treatises. Trans. by Y. R. Simon, J. J. Glanville and G. D. Hollenhorst. Chicago: The University of Chicago Press, 1955.Google Scholar
  30. Ramsey, F. P. ‘Mathematical Logic’. In: Math. Gazette 13.184 (1926), pp. 185–194.CrossRefGoogle Scholar
  31. Dedekind, R. ‘Continuity and Irrational Numbers’. In: Essays on the Theory of Numbers. Trans. by W. W. Beman. New York: Dover, 1901.Google Scholar
  32. Weyl, H. ‘Der circulus vitiosus in der heutigen Begründung der Analysis’. In: Jahresber. d. Deutsch. Math.-Vereinig. 28 (1919), pp. 85–92.Google Scholar
  33. Euclid. Elements. Ed. T. L. Heath. Santa Fe: Green Lion, 2002.Google Scholar
  34. Perlis, A. J. ‘Epigrams on Programming’. In: SIGPLAN Notices 17.9 (1982), pp. 7–13.CrossRefGoogle Scholar
  35. McLarty, C. ‘Poincaré : Mathematics & Logic & Intuition’. In: Phil. Math. 5.2 (1997), pp. 97–115.Google Scholar
  36. Skolem, Th. A. ‘Some remarks on axiomatized set theory’. In: From Frege to Gödel. A source book in mathematical logic, 1879–1931. Ed. by J. van Heijenoort. Cambridge Mass.: Harvard University Press, 1967, pp. 290–301.Google Scholar
  37. Setzer, A. ‘Extending Martin-Löf’s Type Theory by one Mahlo-universe’. In: Arch. Math. Logic 39 (2000), pp. 155–181.CrossRefGoogle Scholar
  38. Husserl, E. Logische Untersuchungen. 3rd ed. Vol. 1. Halle: M. Niemeyer, 1922.Google Scholar
  39. — ‘History of a Fallacy’. In: J. Phil. Ass. 5.19–20 (1958).Google Scholar
  40. Logische Untersuchungen. 5th ed. Vol. 2. Tübingen: M. Niemeyer, 1968.Google Scholar
  41. La géométrie. Paris: A. Hermann, 1886.Google Scholar
  42. Poincaré, H. La Science et l’Hypothèse. Paris: Flammarion, 1902.Google Scholar
  43. Die Grundlagen der Arithmetik. Breslau: W. Koebner, 1884.Google Scholar
  44. — ‘The Arabic Numerals, Numbers and the Definition of Counting’. In: Math. Gazette 40.332 (1956), pp. 114–129.Google Scholar
  45. Recursive number theory. Amsterdam: North-Holland, 1957.Google Scholar

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© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.HorgenSwitzerland

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