Let $k$ be a field. In 1984 Andreas Blass proved that the axiom "for every extension $K|k$, every vector space over $K$ has a basis" implies the axiom of choice. He also raised the question

Does the axiom "every vector space over $k$ has a basis" imply the axiom of choice ?

What's the current status of the question ? Has there been progress ?

1more comments