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

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

The Principle of Open Induction on Cantor space and the Approximate-Fan Theorem

The paper is a contribution to intuitionistic reverse mathematics. We work in a weak formal system for intuitionistic analysis. The Principle of Open Induction on Cantor space is the statement that every open subset of Cantor space that is progressive with respect to the lexicographical ordering of Cantor space coincides with Cantor space. The Approximate-Fan Theorem is an extension of the Fan Theorem that follows from Brouwer's principle of induction on bars in Baire space and implies the Principle of Open Induction on Cantor space. The Principle of Open Induction in Cantor space implies the Fan Theorem, but, conversely the Fan Theorem does not prove the Principle of Open Induction on Cantor space. We list a number of equivalents of the Principle of Open Induction on Cantor space and also a number of equivalents of the Approximate-Fan Theorem

The Fan Theorem, its strong negation, and the determinacy of games

IIn the context of a weak formal theory called Basic Intuitionistic Mathematics $\mathsf{BIM}$, we study Brouwer's Fan Theorem and a strong negation of the Fan Theorem, Kleene's Alternative (to the Fan Theorem). We prove that the Fan Theorem is equivalent to contrapositions of a number of intuitionistically accepted axioms of countable choice and that Kleene's Alternative is equivalent to strong negations of these statements. We also discuss finite and infinite games and introduce a constructively useful notion of determinacy. We prove that the Fan Theorem is equivalent to the Intuitionistic Determinacy Theorem, saying that every subset of Cantor space is, in our constructively meaningful sense, determinate, and show that Kleene's Alternative is equivalent to a strong negation of a special case of this theorem. We then consider a uniform intermediate value theorem and a compactness theorem for classical propositional logic, and prove that the Fan Theorem is equivalent to each of these theorems and that Kleene's Alternative is equivalent to strong negations of them. We end with a note on a possibly important statement, provable from principles accepted by Brouwer, that one might call a Strong Fan Theorem.Comment: arXiv admin note: text overlap with arXiv:1106.273

Optimal partitioning of an interval and applications to Sturm-Liouville eigenvalues

We study the optimal partitioning of a (possibly unbounded) interval of the real line into $n$ subintervals in order to minimize the maximum of certain set-functions, under rather general assumptions such as continuity, monotonicity, and a Radon-Nikodym property. We prove existence and uniqueness of a solution to this minimax partition problem, showing that the values of the set-functions on the intervals of any optimal partition must coincide. We also investigate the asymptotic distribution of the optimal partitions as $n$ tends to infinity. Several examples of set-functions fit in this framework, including measures, weighted distances and eigenvalues. We recover, in particular, some classical results of Sturm-Liouville theory: the asymptotic distribution of the zeros of the eigenfunctions, the asymptotics of the eigenvalues, and the celebrated Weyl law on the asymptotics of the counting function

Reverse Mathematics in Bishop’s Constructive Mathematics

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