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Linear forms and axioms of choice. (English) Zbl 1212.03034

Summary: We work in set theory without choice \(\mathbf {ZF}.\) Given a commutative field \(\mathbb K\), we consider the statement \(\mathbf D (\mathbb K)\): “On every non-null \(\mathbb K\)-vector space there exists a non-null linear form.” We investigate various statements which are equivalent to \(\mathbf D (\mathbb K)\) in \(\mathbf {ZF}.\) Denoting by \(\mathbb Z_2\) the two-element field, we deduce that \(\mathbf D (\mathbb Z_2)\) implies the axiom of choice for pairs. We also deduce that \(\mathbf D (\mathbb Q)\) implies the axiom of choice for linearly ordered sets isomorphic with \(\mathbb Z\).

MSC:

03E25 Axiom of choice and related propositions
15A03 Vector spaces, linear dependence, rank, lineability
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