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

Abstract

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.