The Vector Space Kinna-Wagner Principle is Equivalent to the Axiom of Choice

Abstract

We show that the axiom of choice AC is equivalent to the Vector Space Kinna-Wagner Principle, i.e., the assertion: “For every family 𝒱= {Vi : i ∈ k} of non trivial vector spaces there is a family ℱ = {Fi : i ∈ k} such that for each i ∈ k, Fi is a non empty independent subset of Vi”. We also show that the statement “every vector space over ℚ has a basis” implies that every infinite well ordered set of pairs has an infinite subset with a choice set, a fact which is known not to be a consequence of the axiom of multiple choice MC.