Constructive aspects of Riemann’s permutation theorem for series

Abstract

Abstract The notions of permutable and weak-permutable convergence of a series $\sum _{n=1}^{\infty }a_{n}$ of real numbers are introduced. Classically, these two notions are equivalent, and, by Riemann’s two main theorems on the convergence of series, a convergent series is permutably convergent if and only if it is absolutely convergent. Working within Bishop-style constructive mathematics, we prove that Ishihara’s principle BD- $\mathbb {N}$ implies that every permutably convergent series is absolutely convergent. Since there are models of constructive mathematics in which the Riemann permutation theorem for series holds but BD- $\mathbb{N}$ does not, the best we can hope for as a partial converse to our first theorem is that the absolute convergence of series with a permutability property classically equivalent to that of Riemann implies BD- $\mathbb {N}$ . We show that this is the case when the property is weak-permutable convergence.