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 A = { A i : i ∈ k } \mathcal {A}=\{A_i:i\in k\} of non empty sets there exists a family F = { F i : i ∈ k } \mathcal {F=}\{F_i:i\in k\} of odd sized sets such that, for every i ∈ k i\in k , F i ⊆ A F_i\subseteq A iff in every vector space B B over the two-element field every subspace V ⊆ B V\subseteq B has a complementary subspace S S iff every quotient group of an abelian group each of whose elements has order 2 has a set of representatives.