Axiom of dependent choice

The axiom of dependent choice (from English axiom of dependent choice or principle of dependent choice for short DC) is an axiom of set theory . It is a weak version of the axiom of choice , but it is sufficient in analysis , for example , to show the equivalence of continuity and continuity of sequences . From the axiom the countable axiom of choice follows , but it is weaker than the full axiom of choice. In descriptive set theory it is sometimes used as a substitute for the axiom of choice. It is also called the principle of dependent elections .

The axiom was formulated by Paul Bernays in 1942.

Formal description

Let be a non-empty set and a definite relation . Then there is a sequence in such a way that holds. ${\ displaystyle X}$ ${\ displaystyle R \ subseteq X ^ {2}}$ ${\ displaystyle \ left (x_ {n} \ right) _ {n \ in \ mathbb {N}}}$ ${\ displaystyle X}$ ${\ displaystyle \ forall n \ in \ mathbb {N} \ colon R \ left (x_ {n}, x_ {n + 1} \ right)}$

Even without a dependent selection, a finite initial segment of any length of such a sequence can be formed; the dependent selection thus provides the statement that an infinite sequence can be formed in this way.

The dependent selection only for the real numbers, i.e. only for , is denoted by DC _R. ${\ displaystyle X = \ mathbb {R}}$

use

The axiom of dependent choice is a sufficient fragment of the axiom of choice to construct a sequence using countable transfinite recursion . If it is necessary to make a selection from an infinite number of steps, this may not be possible without the axiom.

Equivalent statements

In the theory ZF , the axiom of dependent choice is equivalent to Baire's theorem in complete metric spaces . In addition, every non-empty tree without leaves has a branch.

Relation to other axioms

Dependent choice is not enough to prove the existence of a subset of the real numbers that is not measurable or does not have the Baire property . However, this is possible with the axiom of choice in its full strength.

The axiom of dependent choice implies the countable axiom of choice, the reverse is not true.

Like the axiom of choice, the axiom of dependent choice is independent of ZF .

Individual evidence

↑ Jürgen Elstrodt: Measure and Integration Theory (2010) p. 98
^ Charles E. Blair: The Baire category theorem implies the principle of dependent choices. In: Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronomer. Phys. 25 (1977), No. 10, pp. 933-934.

[1] Jürgen Elstrodt: Measure and Integration Theory (2010) p. 98

[2] Charles E. Blair: The Baire category theorem implies the principle of dependent choices. In: Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronomer. Phys. 25 (1977), No. 10, pp. 933-934.