A. Pelc, On some weak forms of the axiom of choice in set theory,
*Bulletin de l'Academie Polonaise des Sciences,
Serie des Sciences math., astr. et phys.*
Vol.26, No.7 (1978), pp. 585-589.

Abstract:

The Dedekind sets and their generalisations are considered. It is shown which weak forms of AC are true in the class *WO ^{infinity}* and some consequences of the sentence ``There does not exist a decreasing sequence of cardinalities'' are proved.