In mathematics, the axiom of dependent choices, denoted DC, is a weak form of the axiom of choice (AC) which is still sufficient to develop most of real analysis. 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.
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 choices 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.
DC is (over the theory ZF) equivalent to the statement that every (nonempty) pruned tree has a branch. It is also equivalent to the Baire category theorem for complete metric spaces.
The axiom of dependent choice implies the Axiom of countable choice, and is strictly stronger.
Famous quotes containing the words axiom of, axiom, dependent and/or choice:
“It’s an old axiom of mine: marry your enemies and behead your friends.”
—Robert N. Lee. Rowland V. Lee. King Edward IV (Ian Hunter)
“It’s an old axiom of mine: marry your enemies and behead your friends.”
—Robert N. Lee. Rowland V. Lee. King Edward IV (Ian Hunter)
“For my part, I have no hesitation in saying that although the American woman never leaves her domestic sphere and is in some respects very dependent within it, nowhere does she enjoy a higher station . . . if anyone asks me what I think the chief cause of the extraordinary prosperity and growing power of this nation, I should answer that it is due to the superiority of their woman.”
—Alexis de Tocqueville (1805–1859)
“Then did they strive with emulation who should repeat most wise maxims importing the necessity of suspicion in the choice of our friends—such as “mistrust is the mother of security,” with many more to the same effect.... But notwithstanding the esteem which they professed for suspicion, yet did they think proper to veil it under the name of caution.”
—Sarah Fielding (1710–1768)