Abstract
In presence of continuous choice the fan theorem is equivalent to each pointwise continuous function f from the Cantor space to the natural numbers being uniformly continuous. We investigate whether we can prove this equivalence without the use of continuous choice. By strengthening the assumption of pointwise continuity of f to the assertion that f has a modulus of pointwise continuity which itself is pointwise continuous, we obtain the desired equivalence.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aczel, P., Rathjen, M.: Notes on Constructive Set Theory. Report No. 40, Institut Mittag-Leffler, The Royal Swedish Academy of Sciences (2000/2001)
Bishop, E., Bridges, D.: Constructive Analysis. In: Graph Reduction 1986, vol. 279, Springer, Heidelberg (1967)
Loeb, I.: Equivalents of the (Weak) Fan Theorem. Annals of Pure and Applied Logic 132(1), 51–66 (2005)
Bridges, D., Richman, F.: Varieties of Constructive Mathematics. Cambridge University Press, Cambridge (1987)
Ishihara, H.: Constructive Reverse Mathematics: Compactness Properties. In: Crosilla, L., Schuster, P. (eds.) From Sets and Types to Topology and Analysis, Oxford University Press, Oxford (to appear)
Ishihara, H.: Informal constructive reverse mathematics. In: Feasibility of theoretical arguments of mathematical analysis on computer (Japanese) (Kyoto 2003) (2003); Sūrikaisekikenkyūsho Kōkyūroku 1381, 108–117 (2004)
Moerdijk, I.: Heine-Borel Does not Imply the Fan Theorem. J. Symb. Log. 49(2), 514–519 (1984)
Troelstra, A.S., van Dalen, D.: Constructivism in Mathematics. An Introduction, vol. 1. North-Holland Publ. Co., Amsterdam (1988)
Veldman, W.: The Fan Theorem as an Axiom (preprint)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Berger, J. (2005). The Fan Theorem and Uniform Continuity. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds) New Computational Paradigms. CiE 2005. Lecture Notes in Computer Science, vol 3526. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11494645_3
Download citation
DOI: https://doi.org/10.1007/11494645_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-26179-7
Online ISBN: 978-3-540-32266-5
eBook Packages: Computer ScienceComputer Science (R0)