Axiom of dependent choice
The axiom can be stated as follows: For any nonempty set X and any entire binary relation R on X, there is a sequence (xn) in X such that xnRxn+1 for each n in N. (Here an entire binary relation on X is one such that for each a in X there is a b in X such that aRb.) Note that even without such an axiom we could form the first n terms of such a sequence, for any natural number n; the axiom of dependent choice merely says that we can form a whole sequence this way.
If the set X above is restricted to be the set of all real numbers, the resulting axiom is called DCR.
DC is the fragment of AC required to show the existence of a sequence constructed by transfinite recursion of countable length, if it is necessary to make a choice at each step and if some of those choices cannot be made independently of previous choices.
DC is (over the theory ZF) equivalent to the statement that every (nonempty) pruned tree with ω levels has a branch.
DC also is (over the theory ZF) equivalent to the statement "Lowenheim-Skolem Theorem" - Statement: 'if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ.'
Relation with other axioms
The axiom of dependent choice implies the Axiom of countable choice, and is strictly stronger.
- Blair, Charles E. The Baire category theorem implies the principle of dependent choices. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), no. 10, 933--934.