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

TL;DR

This paper explores the equivalences of Brouwer's Fan Theorem and Kleene's Alternative within a formal system for intuitionistic reverse mathematics, analyzing their implications for compactness and noncompactness in Baire space and real numbers.

Contribution

It introduces BIM, a formal system for intuitionistic reverse mathematics, and identifies numerous equivalents of the Fan Theorem and Kleene's Alternative, highlighting their symmetry and differences.

Findings

Characterization of compact and noncompact closed-and-separable sets

Many equivalent statements for the Fan Theorem and Kleene's Alternative

Analysis of compactness in Baire space and the real interval

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 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 and of the set of the real numbers. Such sets may be compact and also positively noncompact. The Fan Theorem is the statement that Cantor space, or, equivalently, the unit interval, is compact, and Kleene's Alternative is the statement that Cantor space, or, equivalently, the unit interval 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.