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Constructive decidability of classical continuity


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.

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A. Bauer and D. Lešnik (2012) Metric spaces in synthetic topology. Annals of Pure and Applied Logic 163 (2) 87100.

M. Beeson (1985) Foundations of Constructive Mathematics, Springer.

D. Bridges and F. Richman (1987) Varieties of Constructive Mathematics, London Mathematical Society Lecture Note Series volume 97, Cambridge University Press, Cambridge.

A. S. Troelstra and D. van Dalen (1988) Constructivism in Mathematics: An Introduction, Studies in Logic and the Foundations of Mathematics volume 121 and 123, North Holland, Amsterdam.

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Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
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