In mathematics, the axiom of dependent choice, is a weak form of the axiom of choice) that is still sufficient to develop much of real analysis. It was introduced by Paul Bernays in a 1942 article in reverse mathematics that explores which set-theoretic axioms are needed to develop analysis.

Formal statement

A homogeneous relation is called a total relation.

The axiom of dependent choice can be stated as follows: For every nonempty set there exists a sequence such that

, together with the total relation on this set of the second sequence being obtained from the first by appending a single term.)

above is restricted to be the set of all real numbers

Use

terms of such a sequence. The axiom of dependent choice says that we can form a whole (countably infinite) sequence this way.

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

(Zermelo–Fraenkel set theory is equivalent to the Baire category theorem for complete metric spaces.

to the downward Löwenheim–Skolem theorem.

to the statement that every pruned tree levels has a branch (proof below).

is equivalent to a weakened form of Zorn's lemma is equivalent to the statement that any partial order such that every well-ordered chain is finite and bounded, must have a maximal element.

Relation with other axioms

) 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, 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.

It is possible to generalize the axiom to produce transfinite sequences. If these are allowed to be arbitrarily long, then it becomes equivalent to the full axiom of choice.

Notes

References