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 $\sum_{n = 0}^\infty a_{p(n)} = r$, and there exists a bijection of the natural numbers $q:\mathbb{N} \cong \mathbb{N}$ such that $\sum_{n = 0}^\infty a_{q(n)}$ diverges.

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 both sets of numbers:

• It is not provable for the Cantor real numbers since not every Cauchy sequence of Cantor real numbers can be proven to converge to a Cantor real number (i.e. the Cantor real numbers are not provably Cauchy complete)
• It 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.

Nevertheless, one could ask about a weaker version of the Riemann series theorem:

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 $\sum_{n = 0}^\infty a_{p(n)} = r$, and there exists a bijection of the natural numbers $q:\mathbb{N} \cong \mathbb{N}$ such that $\sum_{n = 0}^\infty a_{q(n)}$ diverges.

Is this weaker Riemann series theorem true?