Working in ZF (so, no choice): is it possible that there is a set of reals $X$ such that
$\vert X\vert<\mathbb{R}$, but
$X$ generates $\mathbb{R}$ as a subgroup under addition?
This seems weird, but I can't even show that we can't generate $\mathbb{R}$ with a Dedekind-finite set!
This is in the Geometric Set Theory book with Jindra, in particular on pages 190-191 of the version here: https://people.clas.ufl.edu/zapletal/files/balanced14.pdf
The partial order there forces over a Solovay model. The conditions are disjoint pairs of set of reals $(a,b)$, where $a$ is finite and $b$ is countable, with the order of coordinatewise inclusion.
The GST machinery shows that the partial order doesn't add reals. The union of the finite parts of a generic filter will then be a set of reals of cardinality less than the continuum such that, by genericity, every real is a sum of two of them. In fact, the set will be Dedekind-finite.