The Axiom of Choice for Well-Ordered Families and for Familes of Well-Orderable Sets

We show that it is not possible to construct a Fraenkel-Mostowski model in which the axiom of choice for well ordered families of sets and the axiom of choice for sets for well orderable sets are both true, but the axiom of choice is false. We are concerned with the following two consequences of the axiom of choice: C(WO,∞): Every well ordered collection of sets has a choice function. C(∞,WO): Every collection of well orderable sets has a choice function. It is known that C(WO,∞) does not imply the Axiom of Choice (AC) in Zermelo-Fraenkel set theory (ZF) ([4, p 127] and [8]) nor is C(WO,∞) a theorem of ZF (without AC) ([4, p 123 thm 8.3], [5] and [7]). Similarly C(∞,WO) does not imply AC ([6] and [4, p 82, prob. 5.22]) nor is C(∞,WO) a theorem of ZF ([4, prob 7.12]). (In Cohen’s original model of ZF – AC ([1]), in which there is a countable set of generic reals along with a set collecting them, C(∞,WO) is true and C(WO,∞) is false. Mostowski’s linearly ordered Fraenkel-Mostowski (FM) model has this same property. Feferman’s model of ZF – AC ([2]) in which there is a countable set of generic reals, but no set to collect them, has the property that C(WO,∞) is true, but C(∞,WO) is false. An FM model that has this same property is a variation of Fraenkel’s basic model in which the set of atoms, A, is uncountable, the group, G, is the group of all permutations of A, and the filter of subgroups of G is the set of subgroups of G that leave a countable subset of A pointwise fixed.) The question of whether or not C(WO,∞) ∧ C(∞,WO) implies AC is open. Our purpose is to show that there is no FM model in which both C(WO,∞) and C(∞,WO) are true, but AC is false. It has been shown by Howard ([3]) that in every FM model, C(∞,WO) is equivalent to C(∞, < א0), AC for a family of finite sets. In the proof below we use C(∞, < א0) rather than C(∞,WO). Assume that N is an FM model determined by the the set of atoms A, the group G of permutations of A, and the filter Γ of subgroups of G. Assume that C(WO,∞) and C(∞,WO) are true inN and that AC is false. Let X = { f ∈ N : f 1Eastern Michigan University, Ypsilanti, MI and Purdue University, West Lafayette, IN 2Purdue University, West Lafayette, IN