# Axiom of dependent choice

(Redirected from Dependent choice)

In mathematics, the axiom of dependent choice, denoted by ${\displaystyle {\mathsf {DC}}}$, is a weak form of the axiom of choice (${\displaystyle {\mathsf {AC}}}$) that is still sufficient to develop most of real analysis. It was introduced by Bernays (1942).

## Formal statement

The axiom can be stated as follows: For every nonempty set ${\displaystyle X}$ and every entire binary relation ${\displaystyle R}$ on ${\displaystyle X}$, there exists a sequence ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ in ${\displaystyle X}$ such that ${\displaystyle x_{n}Rx_{n+1}}$ for all ${\displaystyle n\in \mathbb {N} }$. (Here, an entire binary relation on ${\displaystyle X}$ is one where for every ${\displaystyle a\in X}$, there exists a ${\displaystyle b\in X}$ such that ${\displaystyle aRb}$.) Note that even without such an axiom, one can use ordinary mathematical induction to form the first ${\displaystyle n}$ terms of such a sequence, for every ${\displaystyle n\in \mathbb {N} }$; the axiom of dependent choice says that we can form a whole sequence this way.

If the set ${\displaystyle X}$ above is restricted to be the set of all real numbers, then the resulting axiom is denoted by ${\displaystyle {\mathsf {DC}}_{\mathbf {R} }}$.

## Use

${\displaystyle {\mathsf {DC}}}$ is the fragment of ${\displaystyle {\mathsf {AC}}}$ that is 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.

## Equivalent statements

${\displaystyle {\mathsf {DC}}}$ is equivalent (over Zermelo–Fraenkel set theory ${\displaystyle {\mathsf {ZF}}}$) to the statement that every pruned tree with ${\displaystyle \omega }$ levels has a branch.

${\displaystyle {\mathsf {DC}}}$ is also equivalent over ${\displaystyle {\mathsf {ZF}}}$ to the Baire category theorem for complete metric spaces.[1]

It is also equivalent over ${\displaystyle {\mathsf {ZF}}}$ to the Löwenheim–Skolem theorem.[2]

## Relation with other axioms

Unlike full ${\displaystyle {\mathsf {AC}}}$, ${\displaystyle {\mathsf {DC}}}$ is insufficient to prove (given ${\displaystyle {\mathsf {ZF}}}$) that there is a non-measurable set of real numbers, or that there is a set of real numbers without the property of Baire or without the perfect set property. This follows because the Solovay model satisfies ${\displaystyle {\mathsf {ZF}}+{\mathsf {DC}}}$, and every set of real numbers in this model is Lebesgue measurable, has the Baire property and has the perfect set property.

The axiom of dependent choice implies the axiom of countable choice and is strictly stronger.[3]

## Footnotes

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.
2. ^ Moore 1982, p. 325. Moore's table states that "Principle of Dependent Choices" ${\displaystyle \rightarrow }$ "Löwenheim–Skolem theorem"—that is, ${\displaystyle {\mathsf {DC}}}$ implies the Löwenheim–Skolem theorem. The converse is proved in Boolos and Jeffrey 1989, pp. 155–156.
3. ^ Moore 1982, p. 325. Moore's table states that "Principle of Dependent Choices" ${\displaystyle \twoheadrightarrow }$ "${\displaystyle {\bf {{C}^{\aleph _{0}}}}}$: Denumerable Axiom"—that is, ${\displaystyle {\mathsf {DC}}}$ implies the axiom of countable choice, but not vice versa (see p. 321 for definition of "${\displaystyle \twoheadrightarrow }$").