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Subjects
03-XX Mathematical logic and foundations
03Exx Set theory
03E25 Axiom of choice and related propositions

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Hausdorffův-Banachův-Tarského paradox

L. E. Dubins (1981)

Pokroky matematiky, fyziky a astronomie

Hilberträume mit amorphen basen

Norbert Brunner (1984)

Compositio Mathematica

Hyperplanes in matroids and the axiom of choice

Marianne Morillon (2022)

Commentationes Mathematicae Universitatis Carolinae

We show that in set theory without the axiom of choice ZF, the statement sH: “Every proper closed subset of a finitary matroid is the intersection of hyperplanes including it” implies AC $^{fin}$ , the axiom of choice for (nonempty) finite sets. We also provide an equivalent of the statement AC $^{fin}$ in terms of “graphic” matroids. Several open questions stay open in ZF, for example: does sH imply the axiom of choice?

Currently displaying 1 – 3 of 3

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