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

Mathematical Logic Quarterly 47 (2):205-210 (2001)

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 [MATHEMATICAL SCRIPT CAPITAL V]= {Vi : i ∈ k} of non trivial vector spaces there is a family ℱ = {Fi : i ∈ k} such that for each i ∈ k, Fiis 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
Keywords Axiom of multiple choice  Axiom of Choice  Kinna‐Wagner principle  Basis of a vectorspace
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DOI 10.1002/1521-3870(200105)47:2<205::AID-MALQ205>3.0.CO;2-I
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