The following question arose from a discussion about the definability of bases of $\mathbb{R}$ as a $\mathbb{Q}$-vector space. 
(ZF without AC) something we can note is that the existence of a (definable) well-ordering of $\mathbb{R}$ is easily seen to be equivalent to that of a (definable) well-ordered basis of $\mathbb{R}$ as a $\mathbb{Q}$-vector space.
After these remarks, the following question seems natural : 
Is it consistent with ZF that there be a basis of $\mathbb{R}$ over $\mathbb{Q}$ that cannot be well-ordered ? 



Relation between well-orderings of $\mathbb{R}$, and bases over $\mathbb{Q}$
The problem is *generally* open. However, recently Liuzhen Wu, Liang Yu, Ralf Schindler and Mariam Beriashvili posted a preprint in which they prove the consistency of the existence of a Hamel basis and $\Bbb R$ cannot be well-ordered. Specifically, they show there is such a basis in Cohen's first model.


This can be found on Ralf's homepage.


