In , the choice axiom is equivalent to the statement that for every vector space there exists basis. So more concretely, consider the question of existence of a Hamel bases of the real numbers over the rational numbers.
The choice axiom implies that for any vector space, including the real numbers considered as a vector space over the rational numbers, there must exist a basis. This is what we are trying to establish through our investigation into the existence of a Hamel basis.