sequences and series - Question on Riemann's rearrangement theorem

Let there be given some conditionally convergent infinite series $S$. Then let $R$ be some real number, and $Q_k$ a rearrangement of $S$ such that the sum is equal to $R$.

Is $Q_k$ unique? In other words, is there some other rearrangement $W_k$ for which the sum of $S$ is also $R$?

I believe the answer is that $Q_k$ is in fact unique, because the cardinality of the set of permutations of the natural numbers (i.e. the positions of each element of the series $S$), is the same as the cardinality of the real numbers; and therefore the mapping from the real numbers to the rearrangements should be $1-1$. Is this line of reasoning valid, or is this result well known?

Answer

The line of reasoning is not valid:

If you find a particular $Q_k$, then I can find infinitely many other rearrangements that give the same value: Pick any finite subset $X$ of $\mathbb{N}$ and let $\sigma$ be any permutation of $X$. Then the re-rearrangement where we start with $Q_k$ and just mix up the subscripts in $X$ according to $\sigma$ will yield the same sum.

The point is that only rearranging a finite number of terms doesn't change the sum.

Your reasoning breaks down where you say "and therefore the mapping should be 1-1". Just because two sets have the same cardinality doesn't mean any function you think of should be 1-1. Having the same cardinality just means "there is some very specific choice of function which is 1-1 and onto" - it says nothing at all about a generic function you pick between two sets.