Set Theory and Its Philosophy: A Critical Introduction

In §9.4 we introduced a principle — the axiom of countable choice — which differed from the axioms of our default theory because it asserted the existence of a set of a particular sort (actually, in this case, a sequence) without supplying a condition that characterizes it uniquely. In this chapter we shall investigate some generalizations of the axiom of countable choice that share this feature, and enquire a little further into whether the lack of uniqueness in such specifications should be regarded as troubling.

Consider the following attempt to prove that a partially ordered set is partially well-ordered iff it contains no strictly decreasing sequences. Certainly one direction of this equivalence is straightforward: the range of a strictly decreasing sequence is a non-empty set with no minimal element. To prove the reverse implication, suppose that A is not partially well-ordered, so that it has a nonempty subset Β without a minimal element. Now choose an element x₀ in Β and define a sequence (x_n) in Β as follows: once x_n has been chosen, let x_n+1 be any element of Β less than x_n. (Such an element exists because Β has no minimal element.) The sequence (x_n) is clearly strictly decreasing.

The difficulty with this argument is that it requires a countable infinity of choices to be made in order to generate the sequence (x_n). However, it cannot be justified by appeal to the axiom of countable choice, because that axiom licenses only independent choices. One way of putting the point might be by appeal to a temporal metaphor: the choices involved are not simultaneous, since x_n+1 cannot be chosen until the value of x_n is known. The necessity for making choices in this way arises sufficiently often in mathematics for it to be worthwhile to single out the set-theoretic principle which licenses the procedure.

Axiom of countable dependent choice. If r is a relation on a set A such that (∀x ∈ Α) (Ǝy ∈ A) (x r y), then for any a ∈ A there exists a sequence (x_n) in A such that x₀ = a and x_n r x_n+1 for all n ∈ ω.

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