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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 2א0” is strictly weaker than AC(R) and (2) each of the statements (i) “if every member of an infinite set of cardinality 2א0 has power 2א0 , then the union has power 2א0” and (ii) “א(2א0 ) = אω” (א(2א0 ) is Hartogs’ aleph, the least א not ≤ 2א0), is strictly weaker than the full axiom of choice AC. Mathematics Subject Classification: 03E25, 03E35, 03E40, 03E45.

Some Weak Forms of the Axiom of Choice Restricted to the Real Line