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Brouwer’s Fan Theorem as an axiom and as a contrast to Kleene’s alternative

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Der Beweis hat eben nicht nur den Zweck, die Wahrheit eines Satzes über jeden Zweifel zu erheben, sondern auch den, eine Einsicht in die Abhängigkeit der Wahrheiten von einander zu gewähren. The aim of the proof is not only to establish the truth of a proposition beyond any doubt but also to make us see how the truths depend on each other. G. Frege, Die Grundlagen der Arithmetik, § 2.

Abstract

The paper is a contribution to intuitionistic reverse mathematics. We introduce a formal system called Basic Intuitionistic Mathematics BIM, and then search for statements that are, over BIM, equivalent to Brouwer’s Fan Theorem or to its positive denial, Kleene’s Alternative to the Fan Theorem. The Fan Theorem is true under the intended intuitionistic interpretation and Kleene’s Alternative is true in the model of BIM consisting of the Turing-computable functions. The task of finding equivalents of Kleene’s Alternative is, intuitionistically, a nontrivial extension of the task of finding equivalents of the Fan Theorem, although there is a certain symmetry in the arguments that we shall try to make transparent. We introduce closed-and-separable subsets of Baire space \({\mathcal{N}}\) and of the set \({\mathcal{R}}\) of the real numbers. Such sets may be compact and also positively noncompact. The Fan Theorem is the statement that Cantor space \({\mathcal{C}}\), or, equivalently, the unit interval [0, 1], is compact and Kleene’s Alternative is the statement that \({\mathcal{C}}\), or, equivalently, [0, 1], is positively noncompact. The class of the compact closed-and-separable sets and also the class of the closed-and-separable sets that are positively noncompact are characterized in many different ways and a host of equivalents of both the Fan Theorem and Kleene’s Alternative is found.

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  1. Faculty of Science, Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, Postbus 9010, 6500 GL, Nijmegen, The Netherlands

    Wim Veldman

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  1. Wim Veldman
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Correspondence to Wim Veldman.

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figure a

It is a common property of axioms and postulates that they do not need any proof or mathematical assurances, but are taken as well-known and become the principles of the things that follow. Procli Diadochi in primum Euclidis Elementorum librum commentarii, B 101.

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Veldman, W. Brouwer’s Fan Theorem as an axiom and as a contrast to Kleene’s alternative. Arch. Math. Logic 53, 621–693 (2014). https://doi.org/10.1007/s00153-014-0384-9

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