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On the set‐theoretic strength of the existence of disjoint cofinal sets in posets without maximal elements
Author(s) -
Howard Paul,
Saveliev Denis I.,
Tachtsis Eleftherios
Publication year - 2016
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201400089
Subject(s) - mathematics , disjoint sets , axiom , element (criminal law) , axiom of choice , combinatorics , set (abstract data type) , maximal element , discrete mathematics , order (exchange) , extensionality , set theory , pure mathematics , geometry , computer science , finance , political science , law , economics , programming language
In set theory without the Axiom of Choice AC , we study the deductive strength of the statements CS (“Every partially ordered set without a maximal element has two disjoint cofinal subsets”), CS ℵ 0(“Every partially ordered set without a maximal element has a countably infinite disjoint family of cofinal subsets”), LCS (“Every linearly ordered set without a maximum element has two disjoint cofinal subsets”), and LCS ℵ 0(“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, CS is equivalent to CS ℵ 0 , LCS + DC (Dependent Choices) implies LCS ℵ 0 , CS does not imply AC in ZFA (Zermelo‐Fraenkel set theory with the Axiom of Extensionality modified in order to allow the existence of atoms), LCS does not imply AC in ZF (Zermelo‐Fraenkel set theory minus AC ) and LCS ℵ 0(hence, LCS ) is strictly weaker than CS in ZFA .