Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-22T15:55:07.979Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  12 March 2014

Hajime Ishihara  and
Satoru Yoshida
Hajime Ishihara
Affiliation:
School of Information Science, Japan Advanced Institute of Science and Technology, Tatsunokuchi, Ishikawa 923-1292, Japan, E-mail: [email protected]
Satoru Yoshida
Affiliation:
School of Information Science, Japan Advanced Institute of Science and Technology, Tatsunokuchi, Ishikawa 923-1292, Japan, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 4 , December 2002 , pp. 1511 - 1519
Copyright
Copyright © Association for Symbolic Logic 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bishop, Errett, Foundations of Constructive Analysis, McGraw-Hill, New York, 1967.Google Scholar
[2]Bishop, Errett and Bridges, Douglas, Constructive analysis, Grundlehren der Mathematischen Wissenschaften, vol. 279 (1985), Springer-Verlag, Heidelberg.Google Scholar
[3]Bridges, Douglas, Constructive Functional Analysis, Pitman, London, 1979.Google Scholar
[4]Bridges, Douglas, Ishihara, Hajime, Schuster, Peter, and Vîţă, Luminiţa, Strong continuity implies uniform sequential continuity, 2001, Preprint.Google Scholar
[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. 83–96.CrossRefGoogle Scholar
[6]Ishihara, Hajime, Continuity properties in constructive mathematics, this Journal, vol. 57 (1992), pp. 557–565.Google Scholar
[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. 5–12.Google Scholar
[8]Troelstra, A. S., Metamathematical Investigation of Intuitionistic Arithmetic and analysis, Springer, Berlin, 1973.CrossRefGoogle Scholar
#x005B;9#x005D;Troelstra, A. S. and van Dalen, D., Constructivism in Mathematics, vol. 1 and 2, North-Holland, Amsterdam, 1988.Google Scholar