Abstract
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.
The main results below were obtained in 1975 while the first author was a visiting fellow at Wolfson College, Oxford, and the second author a fellow of King’s College, Cambridge
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Bibliograpgy
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.
L. E. J. Brouwer, ‘Intuitionistische Zerlegung mathematischer Grundbegriffe’, Jahresberichte der Deutschen Mathematiker Verein 33 (1923) pp 251-256 Also in [3], pp. 275–280.
L. E. J. Brouwer, Collected Works, Vol. 1. ed. A. Heyting, Am terdam 1975
D. van Dalen, ‘Lectures on Intuitionism’, Cambridge Summer School in Mathematical Logic 1971, Springer Lecture Notes, Vol. 337, 1973, pp. 1–94.
D. van Dalen, ‘An interpretation of intuitionistic analysis’, Annals of Mathematical Logic 13 (1978), pp. 1–43.
D. van Dalen and C. E. Gordon, ‘independence problems in sub-systems of intuitionistic arithmetic’, Indagationes Math. 33 (1971), pp. 448–456.
A. Heyting, Intuïtionistische axiomatiek der projectieve meetkunde. Diss., 1925.
A. Heyting, ‘Die Theorie der linearen Gleichungen in eincr Zahlenspezies mit nicht kommutativer Multiplikation’, Math. Annalen 98 (1927 pp 465–490.
A. Heyting, Intuitionism, Amsterdam, 1956.
D. Prawitz; Natural Deduction, Stockholm, 1965.
C. Smorynski, ‘Applications of Kripke models’, in [13].
C. Smorynski, ‘On Axiomatizing Fragments’ (to appear).
A. S Troelstra, Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, Berlin 1973.
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© 1979 D. Reidel Publishing Company, Dordrecht, Holland
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Van Dalen, D., Statman, R. (1979). Equality in the Presence of Apartness. In: Hintikka, J., Niiniluoto, I., Saarinen, E. (eds) Essays on Mathematical and Philosophical Logic. Synthese Library, vol 122. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-9825-4_5
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DOI: https://doi.org/10.1007/978-94-009-9825-4_5
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