Axiom of dependent choice
In mathematics, the axiom of dependent choice, denoted DC, is a weak form of the axiom of choice (AC) that is still sufficient to develop most of real analysis. It was introduced by Bernays (1942).
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.
It is also equivalent to the Baire category theorem for complete metric spaces.
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.'
Unlike full AC, DC is insufficient to prove (given ZF) that there is a nonmeasurable set of reals, or that there is a set of reals without the property of Baire or without the perfect set property.
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