The axiom of choice
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 (xn) in Β as follows: once xn has been chosen, let xn+1 be any element of Β less than xn. (Such an element exists because Β has no minimal element.) The sequence (xn) 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 (xn). 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 xn+1 cannot be chosen until the value of xn 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 (xn) in A such that x0 = a and xn r xn+1 for all n ∈ ω.
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Publication information: Book title: Set Theory and Its Philosophy: A Critical Introduction. Contributors: Michael Potter - Author. Publisher: Oxford University Press. Place of publication: New York. Publication year: 2004. Page number: 238.
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