Bases for vector spaces over the two-element field and the axiom of choice

Abstract:

It is shown that the axiom of choice follows in a weaker form than the Zermelo - Fraenkel set theory from the assertion: every set of generators G of a vector space V over the two element field includes a basis L for V. It is also shown that: for every family $\mathcal {A}=\{A_i:i\in k\}$ of non empty sets there exists a family $\mathcal {F=}\{F_i:i\in k\}$ of odd sized sets such that, for every $i\in k$, $F_i\subseteq A$ iff in every vector space $B$ over the two-element field every subspace $V\subseteq B$ has a complementary subspace $S$ iff every quotient group of an abelian group each of whose elements has order 2 has a set of representatives.