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Variations on a theme by Ishihara

Published online by Cambridge University Press:  12 November 2014

HANNES DIENER
HANNES DIENER*
Affiliation:
School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand Email: hannes.diener@canterbury.ac.nz
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Abstract

Ishihara's tricks have proven to be a highly useful tool in constructive mathematics, since they enable one to make decisions that seem, on first glance, impossible. They do, however, require that one deals with strongly extensional mappings on complete spaces. In this short note, we show how these assumptions can be weakened. Furthermore, we apply these generalizations to give a partial answer to the question, whether constructively we can rule out the existence of injections from Baire space into the natural numbers, to a version of Riemann's per mutation theorem and to a classification problem about cardinalities in constructive reverse mathematics.

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References

Bauer, A. (2014) An injection from Baire space to natural numbers. In this volume.Google Scholar
Berger, J. and Bridges, D.S. (2009) Rearranging series constructively. Journal of Universal Computer Science 15 (17) 3160–3168.Google Scholar
Bishop, E. and Bridges, D. (1985) Constructive Analysis, Springer-Verlag.Google Scholar
Bridges, D. and Mahalanobis, A. (2001) Bounded variation implies regulated: A constructive proof. Journal of Symbolic Logic 66 (4) 1695–1700.Google Scholar
Bridges, D. S. and Vîţă, L. S. (2006) Techniques of Constructive Analysis, Universitext. Springer, New York, 2006.Google Scholar
Berger, J., Bridges, D. S., Diener, H. and Schwichtenberg, H. (2011) Constructive aspects of Riemann's permutation theorem for series. submitted for publication.Google Scholar
Ishihara, H. (1991) Continuity and nondiscontinuity in constructive mathematics. Journal of Symbolic Logic 56 (4) 1349–1354.CrossRefGoogle Scholar
Ishihara, H. (1992) Continuity properties in constructive mathematics. Journal of Symbolic Logic 57 (2) 557–565.CrossRefGoogle Scholar
Lubarsky, R. and Diener, H. (2014) Principles weaker than BD-N. Journal of Symbolic Logic 78 (3) 681–1024.Google Scholar