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Constructive decidability of classical continuity

Published online by Cambridge University Press:  23 December 2014

MARTÍN ESCARDÓ
MARTÍN ESCARDÓ*
Affiliation:
School of Computer Science, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom Email: m.escardo@cs.bham.ac.uk
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Abstract

We show that the following instance of the principle of excluded middle holds: any function on the one-point compactification of the natural numbers with values on the natural numbers is either classically continuous or classically discontinuous. The proof does not require choice and can be understood in any of the usual varieties of constructive mathematics. Classical (dis)continuity is a weakening of the notion of (dis)continuity, where the existential quantifiers are replaced by negated universal quantifiers. We also show that the classical continuity of all functions is equivalent to the negation of the weak limited principle of omniscience. We use this to relate uniform continuity and searchability of the Cantor space.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 25 , Special Issue 7: Computing with Infinite Data: Topological and Logical Foundations Part 1 , October 2015 , pp. 1578 - 1589
Copyright
Copyright © Cambridge University Press 2014 

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References

Bauer, A. and Lešnik, D. (2012) Metric spaces in synthetic topology. Annals of Pure and Applied Logic 163 (2) 87–100.CrossRefGoogle Scholar
Beeson, M. (1985) Foundations of Constructive Mathematics, Springer.CrossRefGoogle Scholar
Berger, U. and Oliva, P. (2006) Modified bar recursion. Mathematical Structures in Computer Science 16 (2) 163–183.Google Scholar
Bishop, E. (1967) Foundations of Constructive Analysis, McGraw-Hill Book Co., New York.Google Scholar
Bridges, D. and Richman, F. (1987) Varieties of Constructive Mathematics, London Mathematical Society Lecture Note Series volume 97, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Diener, H. (2013) Variations on a theme by Ishihara. Mathematical Structures in Computer Science.Google Scholar
Escardó, M. H. (2012) The intrinsic topology of a Martin-Löf universe à la Russell. Available at the author's web page.Google Scholar
Escardó, M. H. (2013) Infinite sets that satisfy the principle of omniscience in any variety of constructive mathematics. Journal of Symbolic Logic To appear.Google Scholar
Escardó, M. H. and Oliva, P. (2010) Bar recursion and products of selection functions. Available from the authors' web page. Available at: http://www.cs.bham.ac.uk/~mhe/ Google Scholar
Ishihara, H. (1991) Continuity and nondiscontinuity in constructive mathematics. Journal of Symbolic Logic 56 (4) 1349–1354.Google Scholar
Troelstra, A. S. and van Dalen, D. (1988) Constructivism in Mathematics: An Introduction, Studies in Logic and the Foundations of Mathematics volume 121 and 123, North Holland, Amsterdam.Google Scholar