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A constructive look at the completeness of the space (ℝ)

  • Hajime Ishihara (a1) and Satoru Yoshida (a2)

Abstract

We show, within the framework of Bishop's constructive mathematics, that (sequential) completeness of the locally convex space (ℝ) of test functions is equivalent to the principle BD-ℕ which holds in classical mathemtatics, Brouwer's intuitionism and Markov's constructive recursive mathematics, but does not hold in Bishop's constructivism.

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[1]Bishop, Errett, Foundations of Constructive Analysis, McGraw-Hill, New York, 1967.
[2]Bishop, Errett and Bridges, Douglas, Constructive analysis, Grundlehren der Mathematischen Wissenschaften, vol. 279 (1985), Springer-Verlag, Heidelberg.
[3]Bridges, Douglas, Constructive Functional Analysis, Pitman, London, 1979.
[4]Bridges, Douglas, Ishihara, Hajime, Schuster, Peter, and Vîţă, Luminiţa, Strong continuity implies uniform sequential continuity, 2001, Preprint.
[5]Goodman, N. and Myhill, J., The formalization of Bishop's constructive mathematics, Toposes, Algebraic Geometry and Logic (Lawvere, F., editor), Springer, Berlin, 1972, pp. 8396.
[6]Ishihara, Hajime, Continuity properties in constructive mathematics, this Journal, vol. 57 (1992), pp. 557565.
[7]Ishihara, Hajime, Sequential continuity in constructive mathematics, Combinatorics, Computability and Logic, Proceedings of the Third International Conference on Combinatorics, Computability and Logic, (DMTCS'01) in Constanţa Romania (London) (Calude, CS., Dinneen, M.J., and Sburlan, S., editors), Springer-Verlag, 07 2–6, 2001, pp. 512.
[8]Troelstra, A. S., Metamathematical Investigation of Intuitionistic Arithmetic and analysis, Springer, Berlin, 1973.
#x005B;9#x005D;Troelstra, A. S. and van Dalen, D., Constructivism in Mathematics, vol. 1 and 2, North-Holland, Amsterdam, 1988.

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A constructive look at the completeness of the space (ℝ)

  • Hajime Ishihara (a1) and Satoru Yoshida (a2)

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