On the set‐theoretic strength of the existence of disjoint cofinal sets in posets without maximal elements |
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| Authors: | Paul Howard Denis I. Saveliev Eleftherios Tachtsis |
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| Affiliation: | 1. Department of Mathematics, Eastern Michigan University, Ypsilanti, United States of America;2. Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russian Federation;3. Department of Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece |
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| Abstract: | In set theory without the Axiom of Choice  , we study the deductive strength of the statements  (“Every partially ordered set without a maximal element has two disjoint cofinal subsets”),  (“Every partially ordered set without a maximal element has a countably infinite disjoint family of cofinal subsets”),  (“Every linearly ordered set without a maximum element has two disjoint cofinal subsets”), and  (“Every linearly ordered set without a maximum element has a countably infinite disjoint family of cofinal subsets”). Among various results, we prove that none of the above statements is provable without using some form of choice,  is equivalent to  ,  +  (Dependent Choices) implies  ,  does not imply  in  (Zermelo‐Fraenkel set theory with the Axiom of Extensionality modified in order to allow the existence of atoms),  does not imply  in  (Zermelo‐Fraenkel set theory minus  ) and  (hence,  ) is strictly weaker than  in  . |
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