Equality in the Presence of Apartness

  • D. Van Dalen
  • R. Statman
Part of the Synthese Library book series (SYLI, volume 122)


The apartness relation was introduced by Brouwer, [1], [2], as a positive analogue of the inequality relation on the continuum. Subsequently Heyting introduced the notion of apartness axiomatically for a treatment of axiomatic geometry and algebra, [7], [8], [9]. In this paper we will take the axiomatic point of view and consider first-order intuitionistic theories of apartness and equality.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    L. E. J. Brouwer, ‘Begründing der Mengenlehre unabhängig vom Logischen Satz Satz vom ausgeschlossenen Dritten. Zweiter Teil: Theorie der Punktmengen’, Koninklijke Nederlandse Akademie van Wetenschappen, Verhandelingen 127 (1919), pp. 3–33. Also in [3], pp. 191–221.Google Scholar
  2. [2]
    L. E. J. Brouwer, ‘Intuitionistische Zerlegung mathematischer Grundbegriffe’, Jahresberichte der Deutschen Mathematiker Verein 33 (1923) pp 251-256 Also in [3], pp. 275–280.Google Scholar
  3. [3]
    L. E. J. Brouwer, Collected Works, Vol. 1. ed. A. Heyting, Am terdam 1975Google Scholar
  4. [4]
    D. van Dalen, ‘Lectures on Intuitionism’, Cambridge Summer School in Mathematical Logic 1971, Springer Lecture Notes, Vol. 337, 1973, pp. 1–94.CrossRefGoogle Scholar
  5. [5]
    D. van Dalen, ‘An interpretation of intuitionistic analysis’, Annals of Mathematical Logic 13 (1978), pp. 1–43.CrossRefGoogle Scholar
  6. [6]
    D. van Dalen and C. E. Gordon, ‘independence problems in sub-systems of intuitionistic arithmetic’, Indagationes Math. 33 (1971), pp. 448–456.Google Scholar
  7. [7]
    A. Heyting, Intuïtionistische axiomatiek der projectieve meetkunde. Diss., 1925.Google Scholar
  8. [8]
    A. Heyting, ‘Die Theorie der linearen Gleichungen in eincr Zahlenspezies mit nicht kommutativer Multiplikation’, Math. Annalen 98 (1927 pp 465–490.CrossRefGoogle Scholar
  9. [9]
    A. Heyting, Intuitionism, Amsterdam, 1956.Google Scholar
  10. [10]
    D. Prawitz; Natural Deduction, Stockholm, 1965.Google Scholar
  11. [11]
    C. Smorynski, ‘Applications of Kripke models’, in [13].Google Scholar
  12. [12]
    C. Smorynski, ‘On Axiomatizing Fragments’ (to appear).Google Scholar
  13. [13]
    A. S Troelstra, Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, Berlin 1973.Google Scholar

Copyright information

© D. Reidel Publishing Company, Dordrecht, Holland 1979

Authors and Affiliations

  • D. Van Dalen
    • 1
  • R. Statman
    • 2
  1. 1.Department of Mathematics and Department of PhilosophyUniversity of UtrechtThe Netherlands
  2. 2.King’s CollegeCambridge UniversityUK

Personalised recommendations