Abstract
Our aim in writing this paper is to present some of the distinctive features of the real number line ℝ as it appears to the constructive mathematician. Throughout this presentation we shall pay particular attention to constructive notions and proofs that differ from their classical counterparts, or whose classical analogues are insubstantial (in the case of notions) or trivial (in the case of proofs). For example, we explain why one classical definition of ‘closed subset of ℝ’ is inappropriate in the constructive setting (6.2); and we devote a considerable amount of space to the property of locatedness, which plays no role whatsoever in traditional analysis (Section 12).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Barwise, J. (Ed.): 1977, Handbook of Mathematical Logic, North-Holland, Amsterdam.
Beeson, M. J.: 1985, Foundations of Constructive Mathematics, Springer-Verlag, Berlin.
Bishop, Errett: 1967, Foundations of Constructive Analysis, McGraw-Hill, New York.
Bishop, Errett: 1970, ‘Mathematics as a numerical language’, in A. Kino, J. Myhill and R. Vesley (Eds.), Intuitionism and Proof Theory, North-Holland, Amsterdam.
Bishop, Errett and Bridges, Douglas: 1985, Constructive Analysis, Grundlehren der math. Wissenschaften, Bd 279, Springer-Verlag, Berlin.
Bridges, D.S.: 1979, ‘Connectivity properties of metric spaces’, Pacific J. Math., 80(2), 325–331.
Bridges, Douglas and Richman, Fred: 1987, Varieties of Constructive Mathematics, London. Math. Soc. Lecture Notes, 97, Cambridge Univ. Press.
Brouwer, L. E. J.: 1981, Brouwer’s Cambridge Lectures on Intuitionism, Dirk van Dalen (Ed.), Cambridge University Press.
Dieudonné, J.: 1960, Foundations of Modern Analysis, Academic Press.
Dummett, Michael: 1977, Elements of Intuitionism, Oxford University Press.
Goodman, N. and Myhill, J.: 1978, ‘Choice implies excluded middle’, Z. Math. Logik Grundlagen Math., 23, 461.
Heyting, A.: 1971, Intuitionism, 3rd edn., North-Holland, Amsterdam.
Kleene, S. C, and Vesley, R. E.: 1965, The Foundations of Intuitionistic Mathematics, North-Holland, Amsterdam.
Ko, Ker-i: 1991, Complexity Theory of Real Functions, Birkhaüser, Boston.
Kushner, B. A.: 1985, Lectures on Constructive Mathematical Analysis, American Mathematical Society, Providence, R.I.
Lifschitz, V.: 1982, ‘Constructive assertions in an extension of classical mathematics’, J. Symbolic Logic, 47, 359–387.
Mandelkern, M.: 1981, ‘Located sets on the line’, Pacific J. Math., 95, 401–409.
Mandelkern, M.: 1982, ‘Components of an open set’, J. Austral. Math. Soc. (Series A), 33, 249–261.
Mandelkern, M.: 1983, ‘Constructive Continuity’, Memoirs of the Amer. Math. Soc, 277.
Rogers, Hartley: 1967, Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York.
Shanin, N. A.: 1963, ‘On the constructive interpretation of mathematical judgments’, Amer. Math. Soc. Translations, Series 2, 23, 109–189.
Specker, E.: 1949, ‘Nicht konstruktiv beweisbare Sätze der Analysis’, J. Symbolic Logic 14, 145–148.
Staples, J.: 1971, ‘On constructive fields’, Proc. London Math. Soc., 23, 753–768.
Troelstra, A. S.: 1980, ‘Intuitionistic extensions of the reals’, Nieuw Archief voor Wiskunde (3), XXVIII, 63–113.
Troelstra, A. S.: 1982, ‘Intuitionistic extensions of the reals II’, in D. van Dalen, D. Lascar, and J. Smiley, (Eds.), Logic Colloquium ‘80, North-Holland, pp. 279–310.
Troelstra, A. S., and van Dalen, D.: 1988, 1989, Constructivism in Mathematics, North- Holland, Amsterdam, (Vol. I) and (Vol. II).
van Dalen, D.: 1982, ‘Braucht die konstruktive Mathematik Grundlagen?’, Jahrber. Deutsch. Math-Verein, 84, 57–78.
Weyl, H.: 1966, Das Kontinuum, Chelsea Publ. Co., New York.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Bridges, D.S. (1994). A Constructive Look at the Real Number Line. In: Ehrlich, P. (eds) Real Numbers, Generalizations of the Reals, and Theories of Continua. Synthese Library, vol 242. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8248-3_2
Download citation
DOI: https://doi.org/10.1007/978-94-015-8248-3_2
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4362-7
Online ISBN: 978-94-015-8248-3
eBook Packages: Springer Book Archive