Assuming the axiom of choice does not hold we have that there is a vector space without a basis. The situation can be, in some sense, worse. It is consistent that there are vector spaces that have two bases with completely different cardinalities.
Is anything known on when a vector space is spanned by sets of different cardinalities, and on the relation between those cardinalities?
Is there a known relation between common choice principles (BPIT, DC, etc.) and possible cardinalities of a vector space? (For example, does BPIT implies that every two bases have the same size?)