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ON SOME OF BROUWER’S AXIOMS

Part of: Proof theory and constructive mathematics Set theory

Published online by Cambridge University Press:  16 April 2025

WIM VELDMAN Open the ORCID record for WIM VELDMAN [Opens in a new window]
WIM VELDMAN*
Affiliation:
INSTITUTE FOR MATHEMATICS ASTROPHYSICS AND PARTICLE PHYSICS FACULTY OF SCIENCE RADBOUD UNIVERSITY POSTBUS 9010 6500 GL NIJMEGEN THE NETHERLANDS E-mail: [email protected]
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Abstract

We argue that some of Brouwer’s assumptions, rejected by Bishop, should be considered and studied as possible axioms. We show that Brouwer’s Continuity Principle enables one to prove an intuitionistic Borel Hierarchy Theorem. We also explain that Brouwer’s Fan Theorem is useful for a development of the theory of measure and integral different from the one worked out by Bishop. We show that Brouwer’s bar theorem not only proves the Fan Theorem but also a stronger statement that we call the Almost-fan Theorem. The Almost-fan Theorem implies intuitionistic versions of Ramsey’s Theorem and the Bolzano-Weierstrass Theorem.

Keywords

Intuitionistic mathematicsContinuity PrincipleFan TheoremBar TheoremAlmost-Fan TheoremBorel Hierarchy Theoremconstructive measure theoryRamsey’s theorem

MSC classification

Primary: 03E15: Descriptive set theory 03F55: Intuitionistic mathematics 03F60: Constructive and recursive analysis
Type
Article
Information
Bulletin of Symbolic Logic , Volume 31 , Issue 1 , March 2025 , pp. 1 - 52
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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Footnotes

†

Sadly, the author passed away on November 30, 2024.

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