Skip to main content

The Notion of Set

  • Chapter
  • First Online:
  • 915 Accesses

Part of the book series: Logic, Epistemology, and the Unity of Science ((LEUS,volume 22))

Abstract

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 .

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   119.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   159.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   159.99
Price excludes VAT (USA)
  • Durable hardcover 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  • Troelstra, A. S. Principles of Intuitionism. Springer, 1969.

    Google Scholar 

  • De Morgan, A. ‘On the Structure of the Syllogism’. In: Trans. Cambridge Philos. Soc. 8 (1846), pp. 379–408.

    Google Scholar 

  • Symbolic Logic. London-New York: Macmillan & Co., 1896.

    Google Scholar 

  • Analytica posteriora. Trans. by H. Tredennick. Loeb classical library 391. Harvard University Press, 1960, pp. 1–261.

    Google Scholar 

  • Brown, H. C. ‘The Logic of Mr. Russell’. In: J. Philos. Psychol. Sci. Meth. 8.4 (1911), pp. 85–91.

    Google Scholar 

  • — ‘Beiträge zur Begründung der transfiniten Mengenlehre’. In: Math. Ann. 46.4 (1895), pp. 481–512.

    Google Scholar 

  • Intuitionistic Type Theory. Studies in Proof Theory. Napoli: Bibliopolis, 1984.

    Google Scholar 

  • — ‘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 

  • Cardelli, L. and P. Wegner. ‘On Understanding Types, Data Abstraction, and Polymorphism’. In: ACM Computing Surveys 17.4 (1985), pp. 471–522.

    Article  Google Scholar 

  • Intuitionism : An Introduction. Amsterdam: North-Holland, 1956.

    Google Scholar 

  • Paradoxes of the Infinite. Trans. by D. A. Steele. London: Routledge, 1950.

    Google Scholar 

  • — ‘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 

  • Bishop, E. and D. Bridges. Constructive Analysis. Springer, 1985.

    Google Scholar 

  • Hofmann, M. ‘Extensional concepts in intensional type theory’. PhD thesis. LFCS Edinburgh, 1995.

    Google Scholar 

  • Boole, G. ‘The Calculus of Logic’. In: Cambridge and Dublin Math. J. 3 (1848), pp. 183–198.

    Google Scholar 

  • Cantor, G. ‘Über unendliche, lineare Punktmannichfaltigkeiten’. In: Math.Ann. 20.1 (1882), pp. 113–121.

    Article  Google Scholar 

  • — ‘The Nature and Meaning of Numbers’. In: Essays on the Theory of Numbers. Trans. by W. W. Beman. New York: Dover, 1901.

    Google Scholar 

  • — ‘Untersuchungen über die Grundlagen der Mengenlehre—I’. In: Math. Ann. 65 (1908), pp. 261–281.

    Google Scholar 

  • Peano, G. Arithmetices Principia Nova Methodo Exposita. Turin: Fratelli Bocca, 1889.

    Google Scholar 

  • Whitehead, A. N. and B. Russell. Principia Mathematica. Vol. 1. Cambridge University Press, 1910.

    Google Scholar 

  • — ‘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 

  • Peirce, C. S. ‘On the Algebra of Logic’. In: Amer. J. Math. 3.1 (1880), pp. 15–57.

    Article  Google Scholar 

  • Metaphysics. Trans. by H. Tredennick. Loeb Classical Library 271 & 287. Harvard University Press, 1933 & 1935.

    Google Scholar 

  • In duodecim libros Metaphysicorum Aristotelis expositio. Ed. by M.-R. Cathala and R. M. Spiazzi. Turin: Marietti, 1950.

    Google Scholar 

  • Frege, G. Begriffsschrift. Halle: Louis Nebert, 1879.

    Google Scholar 

  • — ‘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 

  • — ‘Mathematical Logic as Based on the Theory of Types’. In: Amer. J. Math. 30.3 (1908), pp. 222–262.

    Google Scholar 

  • Brouwer, L. E. J. ‘Intuitionism and formalism’. Trans. by A. Dresden. In: Bull.Amer. Math. Soc. 20.2 (1913), pp. 81–96.

    Article  Google Scholar 

  • 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 

  • Ramsey, F. P. ‘Mathematical Logic’. In: Math. Gazette 13.184 (1926), pp. 185–194.

    Article  Google Scholar 

  • Dedekind, R. ‘Continuity and Irrational Numbers’. In: Essays on the Theory of Numbers. Trans. by W. W. Beman. New York: Dover, 1901.

    Google Scholar 

  • Weyl, H. ‘Der circulus vitiosus in der heutigen Begründung der Analysis’. In: Jahresber. d. Deutsch. Math.-Vereinig. 28 (1919), pp. 85–92.

    Google Scholar 

  • Euclid. Elements. Ed. T. L. Heath. Santa Fe: Green Lion, 2002.

    Google Scholar 

  • Perlis, A. J. ‘Epigrams on Programming’. In: SIGPLAN Notices 17.9 (1982), pp. 7–13.

    Article  Google Scholar 

  • McLarty, C. ‘Poincaré : Mathematics & Logic & Intuition’. In: Phil. Math. 5.2 (1997), pp. 97–115.

    Google Scholar 

  • 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 

  • Setzer, A. ‘Extending Martin-Löf’s Type Theory by one Mahlo-universe’. In: Arch. Math. Logic 39 (2000), pp. 155–181.

    Article  Google Scholar 

  • Husserl, E. Logische Untersuchungen. 3rd ed. Vol. 1. Halle: M. Niemeyer, 1922.

    Google Scholar 

  • — ‘History of a Fallacy’. In: J. Phil. Ass. 5.19–20 (1958).

    Google Scholar 

  • Logische Untersuchungen. 5th ed. Vol. 2. Tübingen: M. Niemeyer, 1968.

    Google Scholar 

  • La géométrie. Paris: A. Hermann, 1886.

    Google Scholar 

  • Poincaré, H. La Science et l’Hypothèse. Paris: Flammarion, 1902.

    Google Scholar 

  • Die Grundlagen der Arithmetik. Breslau: W. Koebner, 1884.

    Google Scholar 

  • — ‘The Arabic Numerals, Numbers and the Definition of Counting’. In: Math. Gazette 40.332 (1956), pp. 114–129.

    Google Scholar 

  • Recursive number theory. Amsterdam: North-Holland, 1957.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Johan Georg Granström .

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media B.V.

About this chapter

Cite this chapter

Granström, J.G. (2011). The Notion of Set. In: Treatise on Intuitionistic Type Theory. Logic, Epistemology, and the Unity of Science, vol 22. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1736-7_3

Download citation

Publish with us

Policies and ethics