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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]
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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

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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