If you accept that _all_ vector spaces have a (Hamel) basis, you can then prove the Axiom of Choice: http://www.math.lsa.umich.edu/~ablass/bases-AC.pdf http://w

If you accept that _all_ vector spaces have a (Hamel) basis, you can then prove the Axiom of Choice: http://www.math.lsa.umich.edu/~ablass/bases-AC.pdf http://www.math.lsa.umich.edu/~ablass/bases-AC.pdf This means if you want to deny the Axiom in some cases, you will also have to allow for the existence of vector spaces without a basis.