# 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).

## Contents

## Formal statement

The axiom can be stated as follows: For any nonempty set *X* and any entire binary relation *R* on *X*, there is a sequence (*x*_{n}) in *X* such that *x*_{n}*R**x*_{n+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 **DC _{R}**.

## Use

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

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

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

- ↑ 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

- Bernays, Paul (1942), "A system of axiomatic set theory. III. Infinity and enumerability. Analysis.",
*J. Symbolic Logic*,**7**: 65–89, MR 0006333<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - Jech, Thomas, 2003.
*Set Theory: The Third Millennium Edition, Revised and Expanded*. Springer. ISBN 3-540-44085-2.