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Some Weak Forms of the Axiom of Choice Restricted to the Real Line
Author(s) -
Keremedis Kyriakos,
Tachtsis Eleftherios
Publication year - 2001
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 - 10.1002/1521-3870(200108)47:3<413::aid-malq413>3.0.co;2-4
Subject(s) - axiom of choice , mathematics , cardinality (data modeling) , countable set , zermelo–fraenkel set theory , statement (logic) , urelement , real line , axiom , line (geometry) , combinatorics , choice function , power set , discrete mathematics , set (abstract data type) , mathematical economics , set theory , computer science , geometry , political science , law , data mining , programming language
It is shown that AC(ℝ), the axiom of choice for families of non‐empty subsets of the real line ℝ, does not imply the statement PW(ℝ), the powerset of ℝ can be well ordered. It is also shown that (1) the statement “the set of all denumerable subsets of ℝ has size 2 ℵ 0” is strictly weaker than AC(ℝ) and (2) each of the statements (i) “if every member of an infinite set of cardinality 2 ℵ 0has power 2 ℵ 0, then the union has power 2 ℵ 0” and (ii) “ℵ(2 ℵ 0) ≠ ℵ ω ” (ℵ(2 ℵ 0) is Hartogs' aleph, the least ℵ not ≤ 2 ℵ 0), is strictly weaker than the full axiom of choice AC.