[Paper Review] Constructive aspects of Riemann's permutation theorem for series

Josef Berger, Douglas Bridges+2 more|arXiv (Cornell University)|Mar 28, 2013

Computability, Logic, AI Algorithms12 references3 citations

TL;DR

This paper investigates constructive versions of Riemann's permutation theorem for series within Bishop-style constructive mathematics (BISH). It introduces the notion of weak-permutable convergence and proves that, over BISH, the absolute convergence of all weak-permutably convergent series implies Ishihara's principle BD-ℕ, while permutable convergence implies absolute convergence only when BD-ℕ is assumed. The key contribution is establishing a constructive equivalence between the absolute convergence of weak-permutable series and BD-ℕ, showing that weak-permutable convergence is strictly weaker than permutable convergence in a constructive setting.

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 \BDN 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 \BDN 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 \BDN. We show that this is the case when the property is weak-permutable convergence.

Motivation & Objective

• To investigate the constructive logical strength of Riemann's permutation theorem for series in Bishop-style constructive mathematics (BISH).
• To introduce and analyze the notion of weak-permutable convergence as a constructive alternative to permutable convergence.
• To determine whether the absolute convergence of weak-permutably convergent series implies Ishihara's principle BD-ℕ.
• To clarify the logical relationship between permutable convergence, weak-permutable convergence, absolute convergence, and BD-ℕ in constructive analysis.

Proposed method

• Introduce the concept of a bracketing of a series, defined by a strictly increasing function f and the block sums b_k = ∑_{i=f(k)}^{f(k+1)-1} a_i.
• Define weak-permutable convergence as convergence of the original series and the existence of a convergent bracketing for every permutation of the series.
• Use a reductio ad absurdum argument to show that if a series is weak-permutably convergent and a permutation leads to a different sum, then a divergent subseries can be constructed.
• Construct a sequence (a_n) of nonnegative rationals from an inhabited, countable, pseudobounded subset S ⊆ ℕ, using properties that ensure convergence and control over partial sums.
• Prove that if such a series ∑a_n is weak-permutably convergent, then S must be bounded, which implies BD-ℕ.
• Use the Diener-Lubarsky model to show that the converse implications cannot be reversed, establishing strict logical separation between the principles.

Experimental results

Research questions

• RQ1 Does the absolute convergence of every permutably convergent series imply Ishihara’s principle BD-ℕ within BISH?
• RQ2 Is weak-permutable convergence constructively equivalent to permutable convergence, or is it strictly weaker?
• RQ3 Can the absolute convergence of weak-permutably convergent series be shown to imply BD-ℕ in a constructive framework?
• RQ4 What is the logical strength of weak-permutable convergence relative to BD-ℕ and absolute convergence in constructive analysis?
• RQ5 Is there a constructive proof of Riemann’s permutation theorem that avoids BD-ℕ, or is BD-ℕ necessary for such results?

Key findings

• Within BISH augmented by BD-ℕ, every permutably convergent series is absolutely convergent.
• The absolute convergence of every weak-permutably convergent series implies BD-ℕ, establishing a constructive equivalence between this convergence property and the principle BD-ℕ.
• Weak-permutable convergence is strictly weaker than permutable convergence in constructive mathematics, as shown by models where the former holds but BD-ℕ fails.
• There is no algorithm that, given a pseudobounded, countable, inhabited subset of ℕ, can prove that the associated weak-permutably convergent series is permutably convergent.
• The classical equivalence between permutable and weak-permutable convergence fails constructively, as the former implies BD-ℕ while the latter does not.
• The Diener-Lubarsky model demonstrates that the implication from absolute convergence of weak-permutably convergent series to BD-ℕ cannot be reversed, confirming the logical independence of the principles.