Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-22T04:15:54.905Z Has data issue: false hasContentIssue false
  • English
  • Français

A continuity principle, a version of Baire's theorem and a boundedness principle

Published online by Cambridge University Press:  12 March 2014

Hajime Ishihara  and
Peter Schuster
Hajime Ishihara
Affiliation:
School of Information Science, Japan Advanced Institute of Science and Technology, Nomi, Ishikawa 923-1292, Japan, E-mail: [email protected]
Peter Schuster
Affiliation:
Mathematisches institut, Universität München, Theresienstraβe 39, 80333 München, Germany, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We deal with a restricted form WC-N′ of the weak continuity principle, a version BT′ of Baire's theorem, and a boundedness principle BD-N. We show, in the spirit of constructive reverse mathematics, that WC-N′, BT′ + ¬LPO and BD-N + ¬LPO are equivalent in a constructive system, where LPO is the limited principle of omniscience.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 4 , December 2008 , pp. 1354 - 1360
Copyright
Copyright © Association for Symbolic Logic 2008

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 mathematics, McGraw-Hill, New York, 1967.Google Scholar
[2]Bishop, Errett and Bridges, Douglas, Constructive analysis, Springer, Berlin, 1985.CrossRefGoogle Scholar
[3]Bridges, Douglas, Ishihara, Hajime, Schuster, Peter, and Vîţă, Luminiţa, Strong continuity implies uniformly sequential continuity. Archive for Mathematical Logic, vol. 44 (2005), pp. 887–895.CrossRefGoogle Scholar
[4]Bridges, Douglas, Ishihara, Hajime, and Vîţă, Luminiţa, A new constructive version of Baire's theorem, Hokkaido Mathematical Journal, vol. 35 (2006), pp. 107–118.CrossRefGoogle Scholar
[5]Bridges, Douglas and Richman, Fred, Varieties of constructive mathematics, London Mathematical Society Lecture Notes Series, vol. 97, Cambridge University Press, London, 1987.CrossRefGoogle Scholar
[6]Bridges, Douglas and Vîţă, Luminiţa, Techniques of constructive analysis, Springer, New York, 2006.Google Scholar
[7]Ishihara, Hajime, Continuity and nondiscontinuity in constructive mathematics, this Journal, vol. 56 (1991), pp. 1349–1354.Google Scholar
[8]Ishihara, Hajime, Continuity properties in constructive mathematics, this Journal, vol. 57 (1992), pp. 557–565.Google Scholar
[9]Ishihara, Hajime, Sequential continuity in constructive mathematics. Combinatorics, computability and logic (Calude, C. S.. Dinneen, M. J., and Sburlan, S., editors). Springer-Verlag, London, 2001, pp. 5–12.CrossRefGoogle Scholar
[10]Ishihara, Hajime, Constructive reverse mathematics: compactness properties, From sets and types to analysis and topology (Crosilla, L. and Schuster, P., editors), Oxford Logic Guides 48, Oxford University Press, 2005, pp. 245–267.CrossRefGoogle Scholar
[11]Ishihara, Hajime, Reverse mathematics in Bishop's constructive mathematics, Philosophia Scientist, vol. 6 (2006), pp. 43–59, Cahier special.CrossRefGoogle Scholar
[12]Ishihara, Hajime and Yoshida, Satoru, A constructive look at the completeness of D(R), this Journal, vol. 67 (2002), pp. 1511–1519.Google Scholar
[13]Lietz, Peter, From constructive mathematics to computable analysis via realizability interpretation, Dissertation, Technische Universität Darmstadt, 2004.Google Scholar
[14]Loeb, Iris, Equivalents of the (weak) fan theorem, Annals of Pure and Applied Logic, vol, 132 (2005), pp. 51–66.CrossRefGoogle Scholar
[15]Negri, Sara and Soravia, Daniele, The continuum as a formal space, Archive for Mathematical Logic, vol. 38 (1999), pp. 423–447.CrossRefGoogle Scholar
[16]Richman, Fred, Inluitionistic notions of boundedness in N, preprint, 2007.Google Scholar
[17]Troelstra, Anne S. and van Dalen, Dirk, Constructivism in mathematics, vol. I and II, NorthHolland, Amsterdam, 1988.Google Scholar
[18]Veldman, Wim, A survey of intuitionistic descriptive set theory, Proceedings of the Heyting conference (Petkov, P. P., editor), Mathematical Logic, Plenum Press, New York and London, 1988, pp. 155–174.Google Scholar
[19]Veldman, Wim, Brouwer's fan theorem as an axiom and as a contrast to Kleene's alternative, preprint, 2005, Radboud University, Nijmegen.Google Scholar