Abstract
Brouwer’s Continuity Principle distinguishes intuitionistic mathematics from other varieties of constructive mathematics, giving it its own flavour. We discuss the plausibility of this assumption and show how it is used. We explain how one may understand its consequences even if one hesitates to accept it as an axiom.
In memory of Johan J. de Iongh (1915–1999)
Trying to learn to use words, and every attempt Is a wholly new start, and a different kind of failure
T.S. Eliot, East Coker, V, 1940
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References
L.E.J. Brouwer (1918), Begründung der Mengenlehre unabhängig vom logischen Satz vorn ausgeschlossenen Dritten. Erster Teil: Allgemeine Mengenlehre. Koninklijke Nederlandsche Akademie van Wetenschappen, Verhandelingen, le sectie 12 no. 5, 43 p. Also: Brouwer 1975, pp. 150–190, with footnotes, pp. 573–585.
L.E.J. Brouwer (1927), Über Definitionsbereiche von Funktionen,Math. Annalen 97, pp. 60–75. (English translation of §§1–3 in: van Heijenoort 1967, pp. 446–463, with an introduction by Charles Parsons, see especially footnote g, page 448).
L.E.J. Brouwer (1975), Collected Works, Vol. I. Philosophy and Foundations of Mathematics, ed. A. Heyting, (North-Holland Publ. Co., Amsterdam etc.).
L.E.J. Brouwer (1991), Intuitionismus,herausgegeben, eingeleitet und kommentiert von D. van Dalen, (Wissenschaftsverlag Mannheim etc.)
J. van Heijenoort (1967), From Frege to Gödel, A Source Book in Mathematical Logic, ( Harvard University Press, Cambridge Mass. )
A. Heyting (1952/53), Inleiding tot de Intuïtionistische Wiskunde,College uitgewerkt door J.J. de Iongh, (Introduction to Intuitionistic Mathematics, Course Notes elaborated by J.J. de Iongh), s.n.
S.C. Kleene and R.E. Vesley (1965), The Foundations of Intuitionistic Mathematics, especially in relation to Recursive Functions, (North-Holland Publ. Co., Amsterdam etc. )
A.S. Troelstra and D. van Dalen (1988), Constructivism in Mathematics, volume I, ( North-Holland Publ. Co., Amsterdam etc. ).
W. Veldman (1981), Investigations in Intuitionistic Hierarchy Theory, Ph.D. Thesis, Katholieke Universiteit Nijmegen.
W. Veldman (1982) On the continuity of functions in intuitionistic real analysis, Report 8210, Department of Mathematics, Katholieke Universiteit Nijmegen.
W. Veldman (1990), A survey of intuitionistic descriptive set theory in P.P. Petkov (ed.), Mathematical Logic, Proceedings of the Heyting Conference 1988, ( Plenum Press, New York and London ), pp. 155–174.
W. Veldman (1995), Some intuitionistic variations on the notion of a finite set of natural numbers, in: H.C.M. de Swart, L.J.M. Bergmans (ed.), Perspectives on Negation, essays in honour of Johan J. de Iongh on the occasion of his 80th birthday, ( Tilburg University Press, Tilburg ), pp. 177–202.
W. Veldman (1999A), On sets enclosed between a set and its double complement, in: A. Cantini e.a. (ed.), Logic and Foundations of Mathematics, Proceedings Xth International Congress on Logic, Methodology and Philosophy of Science, Florence 1995, Volume III, ( Kluwer Academic Publishers, Dordrecht etc. ), pp. 143–154.
W. Veldman (1999B), The Borel hierarchy in intuitionistic mathematics,Report 9930, Department of Mathematics, Katholieke Universiteit Nijmegen, submitted for publication to: Annals of Pure and Applied Logic.
W. Veldman and M. Jansen (1990), Some observations on intuitionistically elementary properties of linear orderings, Archive for Mathematical Logic, vol. 29, pp. 171–187.
W. Veldman and F. Waaldijk (1996), Some elementary results in intuitionistic model theory, Journal of Symbolic Logic, vol. 61, pp. 745–767
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Veldman, W. (2001). Understanding and Using Brouwer’s Continuity Principle. In: Schuster, P., Berger, U., Osswald, H. (eds) Reuniting the Antipodes — Constructive and Nonstandard Views of the Continuum. Synthese Library, vol 306. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9757-9_24
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DOI: https://doi.org/10.1007/978-94-015-9757-9_24
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