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Extensional constructive real analysis via locators

Published online by Cambridge University Press:  02 September 2020

Auke B. Booij Open the ORCID record for Auke B. Booij [Opens in a new window]
Auke B. Booij*
Affiliation:
School of Computer Science, University of Birmingham, Birmingham, UK
*
Email: abb538@cs.bham.ac.uk
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Abstract

Real numbers do not admit an extensional procedure for observing discrete information, such as the first digit of its decimal expansion, because every extensional, computable map from the reals to the integers is constant, as is well known. We overcome this by considering real numbers equipped with additional structure, which we call a locator. With this structure, it is possible, for instance, to construct a signed-digit representation or a Cauchy sequence, and conversely, these intensional representations give rise to a locator. Although the constructions are reminiscent of computable analysis, instead of working with a notion of computability, we simply work constructively to extract observable information, and instead of working with representations, we consider a certain locatedness structure on real numbers.

Keywords

constructive mathematicsconstructive analysishomotopy type theorydependent type theoryexact real arithmetic

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Type
Paper
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Mathematical Structures in Computer Science , Volume 31 , Issue 1 , January 2021 , pp. 64 - 88
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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