The classical Riemann series theorem states that given a sequence $(a_n)_{n \in \mathbb{N}}$ of real numbers such that the series $\sum_{n = 0}^\infty a_n$ is conditionally convergent, for all real numbers $r \in \mathbb{R}$, there exists a bijection of the natural numbers $p:\mathbb{N} \cong \mathbb{N}$ such that $\sum_{n = 0}^\infty a_{p(n)} = r$. There is also a second half of the theorem which states that there exists a bijection of the natural numbers $q:\mathbb{N} \cong \mathbb{N}$ such that $\sum_{n = 0}^\infty a_{q(n)}$ diverges, but for the purposes of this question we shall ignore this part of the Riemann series theorem.
We work in the most general notion of constructive mathematics, where we do not assume any weak choice principles at all. This means that there are multiple notions of real numbers to choose from, including
- the Cantor real numbers $\mathbb{R}_C$, which is defined as a quotient set of Cauchy sequences of rational numbers
- the Dedekind real numbers $\mathbb{R}_D$, which is defined as the set of two-sided Dedekind cuts of the rational numbers
The classical Riemann series theorem is not provable for the Dedekind real numbers since given a sequence $(a_n)_{n \in \mathbb{N}}$ of rational numbers and a bijection of the natural numbers $p:\mathbb{N} \cong \mathbb{N}$, the partial sums of the series $\sum_{n = 0}^\infty a_{p(n)}$ is a sequence of rational numbers, and thus any such series which converges would only converge to the Cantor real numbers. The Dedekind real numbers cannot be proven to be equivalent to the Cantor real numbers.
Since the Cantor real numbers are a subset of the Dedekind real numbers, there are actually two versions of the Riemann series theorem which could be stated for the Cantor real numbers:
- Given a sequence $(a_n)_{n \in \mathbb{N}}$ of Cantor real numbers such that the series $\sum_{n = 0}^\infty a_n$ is conditionally convergent in the Cantor real numbers, for all Cantor real numbers $r \in \mathbb{R}_C$, there exists a bijection of the natural numbers $p:\mathbb{N} \cong \mathbb{N}$ such that $\sum_{n = 0}^\infty a_{p(n)} = r$.
- Given a sequence $(a_n)_{n \in \mathbb{N}}$ of Cantor real numbers such that the series $\sum_{n = 0}^\infty a_n$ is conditionally convergent in the Dedekind real numbers, for all Cantor real numbers $r \in \mathbb{R}_C$, there exists a bijection of the natural numbers $p:\mathbb{N} \cong \mathbb{N}$ such that $\sum_{n = 0}^\infty a_{p(n)} = r$.
These two statements are not the same because the Cantor real numbers cannot be proven to be Cauchy complete; in general, the former is weaker than the latter.
Furthmore, one could ask about the rational numbers
- Given a sequence $(a_n)_{n \in \mathbb{N}}$ of rational numbers such that the series $\sum_{n = 0}^\infty a_n$ is conditionally convergent, for all Cantor real numbers $r \in \mathbb{R}_C$, there exists a bijection of the natural numbers $p:\mathbb{N} \cong \mathbb{N}$ such that $\sum_{n = 0}^\infty a_{p(n)} = r$.
Which of these weaker Riemann series theorems are true?
Edit: Berger and Bridges proved in their paper Rearranging Series Constructively that
- Given a sequence $(a_n)_{n \in \mathbb{N}}$ of rational numbers such that the series $\sum_{n = 0}^\infty a_n$ is conditionally convergent, for all rational numbers $r \in \mathbb{Q}$, there exists a bijection of the natural numbers $p:\mathbb{N} \cong \mathbb{N}$ such that $\sum_{n = 0}^\infty a_{p(n)} = r$.
However, their generalization to the real numbers relies on dependent choice to establish that all the notions of the real numbers are equivalent to each other and the real numbers are Cauchy complete, and so cannot be proven in more general constructive mathematics.
