Axiom of dependent choice

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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.

Formal statement[edit]

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.

Use[edit]

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.

Equivalent statements[edit]

DC is (over the theory ZF) equivalent to the statement that every (nonempty) pruned tree has a branch. It is also equivalent[1] to the Baire category theorem for complete metric spaces.

Relation with other axioms[edit]

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 of dependent choice implies the Axiom of countable choice, and is strictly stronger.

Footnotes[edit]

  1. ^ 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.

References[edit]

  • Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.