Extending Independent Sets to Bases and the Axiom of Choice

Mathematical Logic Quarterly 44 (1):92-98 (1998)
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Abstract

We show that the both assertions “in every vector space B over a finite element field every subspace V ⊆ B has a complementary subspace S” and “for every family [MATHEMATICAL SCRIPT CAPITAL A] of disjoint odd sized sets there exists a subfamily ℱ={Fj:j ϵω} with a choice function” together imply the axiom of choice AC. We also show that AC is equivalent to the statement “in every vector space over ℚ every generating set includes a basis”

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Citations of this work

On vector spaces over specific fields without choice.Paul Howard & Eleftherios Tachtsis - 2013 - Mathematical Logic Quarterly 59 (3):128-146.

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References found in this work

The Axiom of Choice.Thomas J. Jech - 1973 - Amsterdam, Netherlands: North-Holland.
The Axiom of Choice.Gershon Sageev - 1976 - Journal of Symbolic Logic 41 (4):784-785.
Equivalents of the Axiom of Choice, II.Herman Rubin & Jean E. Rubin - 1987 - Journal of Symbolic Logic 52 (3):867-869.

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