Abstract
We prove in set theory without the Axiom of Choice, that Rado’s selection lemma (\({\mathbf{RL}}\)) implies the Hahn-Banach axiom. We also prove that \({\mathbf{RL}}\) is equivalent to several consequences of the Tychonov theorem for compact Hausdorff spaces: in particular, \({\mathbf{RL}}\) implies that every filter on a well orderable set is included in a ultrafilter. In set theory with atoms, the “Multiple Choice” axiom implies \({\mathbf{RL}}\) .
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Morillon, M. Some consequences of Rado’s selection lemma. Arch. Math. Logic 51, 739–749 (2012). https://doi.org/10.1007/s00153-012-0296-5
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DOI: https://doi.org/10.1007/s00153-012-0296-5