nLab Riemann series theorem

 Idea

The set $\Sigma_\mathbb{N} \coloneqq \mathbb{N} \cong \mathbb{N}$ is the symmetric group of the natural numbers, whose elements $p \in \Sigma_\mathbb{N}$ are the permutations of the set of natural numbers.

The Riemann series theorem or the Riemann rearrangement theorem states that given a sequence of real numbers $a:\mathbb{N} \to \mathbb{R}$, if the series $\sum_{n=0}^{\infty} a_n$ is conditionally convergent, then

• for all real numbers $M \in \mathbb{R}$, there exists a permutation $p \in \Sigma_\mathbb{N}$ such that the series $\sum_{n=0}^{\infty} a_{p(n)}$ converges to $M$.

• there exists a permutation $p \in \Sigma_\mathbb{N}$ such that the series $\sum_{n=0}^{\infty} a_{p(n)}$ diverges.

Construction of the real numbers

The Riemann series theorem could be used to construct the real numbers. Since the Riemann series theorem holds for any conditionally convergent series, it suffices to use the conditionally convergent series defined by the sequence of rational numbers $a:\mathbb{N} \to \mathbb{Q}$ where $a_{2 n} = \frac{1}{n}$ and $a_{2 n + 1} = -\frac{1}{n}$. Then we define the real numbers as a quotient set $\mathbb{R} \coloneqq C(\Sigma_\mathbb{N})/\sim$ of the subset of the symmetric group of the natural numbers $C(\Sigma_\mathbb{N}) \subseteq \Sigma_\mathbb{N}$ for which the series $\sum_{n=0}^{\infty} a_{p(n)}$ converges for $p \in \Sigma_\mathbb{N}$; i.e. for which the sequence of partial sums is a Cauchy sequence. We say that permutations $p \in C(\Sigma_\mathbb{N})$ and $q \in C(\Sigma_\mathbb{N})$ are similar $p \sim q$ if their corresponding series have the same limit. This is similar to the construction of the real numbers as the quotient set of all Cauchy sequences of rational numbers.

In constructive mathematics

The Riemann series theorem cannot be proved in constructive mathematics. In general, given a sequence of rational numbers $a:\mathbb{N} \to \mathbb{Q}$ such that the series $\sum_{n=0}^{\infty} a_n$ is conditionally convergent, any permutation of the indices $p \in \Sigma_\mathbb{N}$ such that the series $\sum_{n=0}^{\infty} a_{p(n)}$ converges would only converges to the Cantor real numbers, since the sequence of partial sums of a series of rational numbers is a sequence of rational numbers, and the limit of any Cauchy sequence of rational numbers is only a Cantor real number. In the absence of excluded middle or countable choice, one cannot prove that any sequentially Cauchy complete Archimedean ordered field, such as the Dedekind real numbers, is equivalent to the Cantor real numbers.

 See also

• conditional convergence

 References

• Wikipedia, Riemann series theorem