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

Abstract

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.

DOI: 10.1002/1521-3870(200108)47:3%3C413::AID-MALQ413%3E3.0.CO;2-4

Cite this paper

@article{Keremedis2001SomeWF, title={Some Weak Forms of the Axiom of Choice Restricted to the Real Line}, author={Kyriakos Keremedis and Eleftherios Tachtsis}, journal={Math. Log. Q.}, year={2001}, volume={47}, pages={413-422} }