Axiom of dependent choice explained

In mathematics, the axiom of dependent choice, denoted by

DC , is a weak form of the axiom of choice (AC ) 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.^[1]

Formal statement

R on X is called a total relation if for every a \in X, there exists some b \in X such that aR~b is true.X and every total relation R on X, there exists a sequence (x _{n)} _{n} in X such that x _{n} R~x _{n} for all n \in \N.

In fact, x₀ may be taken to be any desired element of X. (To see this, apply the axiom as stated above to the set of finite sequences that start with x₀ and in which subsequent terms are in relation

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

If the set

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

Use

Even without such an axiom, for any

n , one can use ordinary mathematical induction to form the first n terms of such a sequence.The axiom of dependent choice says that we can form a whole (countably infinite) sequence this way.

The axiom

DC is the fragment of 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

Over

ZF (Zermelo–Fraenkel set theory without the axiom of choice), DC is equivalent to the Baire category theorem for complete metric spaces.^[2]

It is also equivalent over

ZF to the downward Löwenheim–Skolem theorem.^{[3] [4]} DC is also equivalent over ZF to the statement that every pruned tree with \omega levels has a branch (proof below).

Furthermore,

DC is equivalent to a weakened form of Zorn's lemma; specifically DC 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

Unlike full

AC , DC is insufficient to prove (given 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 ZF + 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.^{[5] [6]}

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.

References

• Book: Jech, Thomas . Set Theory . 2003 . Third Millennium . . 3-540-44085-2 . 174929965 . 1007.03002.

Notes and References

1. "The foundation of analysis does not require the full generality of set theory but can be accomplished within a more restricted frame." Bernays . Paul . 1942 . A system of axiomatic set theory . Part III. Infinity and enumerability. Analysis. . . 7 . 2 . 65–89 . 0006333 . 10.2307/2266303 . 2266303. 250344853 . The axiom of dependent choice is stated on p. 86.
2. "The Baire category theorem implies the principle of dependent choices." Blair, Charles E. . 1977 . The Baire category theorem implies the principle of dependent choices . Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. . 25 . 10 . 933–934.
3. Moore states that "Principle of Dependent Choices ⇒ Löwenheim–Skolem theorem" — that is, DC implies the Löwenheim–Skolem theorem. See table Book: Moore, Gregory H. . 1982 . Zermelo's Axiom of Choice: Its origins, development, and influence . 325 . Springer . 0-387-90670-3.
4. The converse is proved in Book: Boolos . George S. . George Boolos . Jeffrey . Richard C. . Richard Jeffrey . 1989 . Computability and Logic . registration . 3rd . 155–156 . Cambridge University Press . 0-521-38026-X .
5. Bernays proved that the axiom of dependent choice implies the axiom of countable choice See esp. p. 86 in Bernays . Paul . 1942 . A system of axiomatic set theory . Part III. Infinity and enumerability. Analysis. . Journal of Symbolic Logic . 7 . 2 . 65–89 . 0006333 . 10.2307/2266303 . 2266303. 250344853 .
6. For a proof that the Axiom of Countable Choice does not imply the Axiom of Dependent Choice see