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. (shrink)

Classically, weak König's lemma and Brouwer's fan theorem for detachable bars are equivalent. We give a direct constructive proof that the former implies the latter.

We will overview the results in an informal approach to constructive reverse mathematics, that is reverse mathematics in Bishop’s constructive mathematics, especially focusing on compactness properties and continuous properties.

This article presents a weak system of intuitionistic second-order arithmetic, WKV, a subsystem of the one in S.C. Kleene, R.E. Vesley [The Foundations of Intuitionistic Mathematics: Especially in Relation to Recursive Functions, North-Holland Publishing Company, Amsterdam, 1965]. It is then shown that some statements of real analysis, like a version of the Heine–Borel Theorem, and some statements of logic, e.g. compactness of classical proposition calculus, are equivalent to the Fan Theorem in this system.