Brouwer's Fan Theorem as an axiom and as a contrast to Kleene's Alternative

Details Brouwer's Fan Theorem as an axiom and as a contrast to Kleene's Alternative Brouwer's Fan Theorem as an axiom and as a contrast to Kleene's Alternative

2011-06-14

arxiv.org/abs/1106.2738v1

Mathematics

Logic

Scientific paper

The paper is a contribution to intuitionistic reverse mathematics. Working in the context of a formal system called Basic Intuitionistic Mathematics BIM, we formulate a large number of equivalents of the Fan Theorem. We introduce the class of the closed-and-separable subsets of Baire space and consider the members of this class that enjoy the so-called Heine-Borel property. The Fan Theorem is the statement that Cantor space has this property. We prove that the class of the closed-and-separable subsets of Baire space with the Heine-Borel-property may be characterized in many different ways. We also consider the class of all closed-and-separable subsets of Baire space that positively fail to have the Heine-Borel-property and show that this class, too, admits of a large number of different descriptions. We then introduce the set of the real numbers and its closed-and-separable subsets, again examining in particular the closed-and-separable sets of reals that have or positively fail to have the Heine-Borel-property. The Fan Theorem is equivalent to the statement: "The unit interval [0,1] has the Heine-Borel property" and, as a corollary of the manifold characterizations of the closed-and-separable subsets of Baire space and of the set of the real numbers, respectively, that have the Heine-Borel-property, we obtain a large number of equivalents of the Fan Theorem. Kleene's Alternative to the Fan Theorem is the statement: "Cantor space positively fails to have the Heine-Borel-property", or, equivalently, "The unit interval [0,1] positively fails to have the Heine-Borel-property", and also this statement turns out to have a large number of equivalent formulations. Note that the Fan Theorem holds in intuitionistic analysis and that Kleene's Alternative is true in the context of intuitionistic recursive analysis.

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