Axiom of dependent choice

Short description: Weak form of the axiom of choice

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 that explores which set-theoretic axioms are needed to develop analysis.[lower-alpha 1]

## Contents

* 1 Formal statement
* 2 Use
* 3 Equivalent statements
* 4 Relation with other axioms
* 5 Notes
* 6 References

## Formal statement

A homogeneous relation R on X is called a total relation if for every a∈X, there exists some b∈X such that aRb is true.

The axiom of dependent choice can be stated as follows: For every nonempty set X and every total relation R on X, there exists a sequence (xn)n∈ℕ in X such that

xnRxn+1 for all n∈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 DCℝ.

## 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.[1]

It is also equivalent over ZF to the downward Löwenheim–Skolem theorem.[lower-alpha 2][2]

DC is also equivalent over ZF to the statement that every pruned tree with ω 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.[3]

Proof that DC⟺ Every pruned tree with ω levels has a branch

(⟸) Let R be an entire binary relation on X. The strategy is to define a tree T on X of finite sequences whose neighboring elements satisfy R. Then a branch through T is an infinite sequence whose neighboring elements satisfy R. Start by defining ⟨x0,…,xn⟩∈T if xkRxk+1 for 0≤k<n. Since R is entire, T is a pruned tree with ω levels. Thus, T has a branch ⟨x0,…,xn,…⟩. So, for all n≥0: ⟨x0,…,xn,xn+1⟩∈T, which implies xnRxn+1. Therefore, DC is true.

(⟹) Let T be a pruned tree on X with ω levels. The strategy is to define a binary relation R on T so that DC produces a sequence tn=⟨x0,…,xf(n)⟩ where tnRtn+1 and f(n) is a strictly increasing function. Then the infinite sequence ⟨x0,…,xk,…⟩ is a branch. (This proof only needs to prove this for f(n)=m+n.) Start by defining uRv if u is an initial subsequence of v,length⁡(u)>0, and length⁡(v)= length⁡(u)+1. Since T is a pruned tree with ω levels, R is entire. Therefore, DC implies that there is an infinite sequence tn such that tnRtn+1. Now t0=⟨x0,…,xm⟩ for some m≥0. Let xm+n be the last element of tn. Then tn=⟨x0,…,xm,…,xm+n⟩. For all k≥0, the sequence ⟨x0,…,xk⟩ belongs to T because it is an initial subsequence of t0(k≤m) or it is a tn(k≥m). Therefore, ⟨x0,…,xk,…⟩ is a branch.

## 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.[4][5]

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

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). "Part III. Infinity and enumerability. Analysis.". Journal of Symbolic Logic. A system of axiomatic set theory 7 (2): 65–89. doi:10.2307/2266303. http://doc.rero.ch/record/290936/files/S0022481200064185.pdf. The axiom of dependent choice is stated on p. 86.
2. ↑ Moore states that "Principle of Dependent Choices ⇒ Löwenheim–Skolem theorem" — that is, DC implies the Löwenheim–Skolem theorem. See table Moore, Gregory H. (1982). Zermelo's Axiom of Choice: Its origins, development, and influence. Springer. p. 325. ISBN 0-387-90670-3.

## References

1. ↑ "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.
2. ↑ The converse is proved in Boolos, George S.; Jeffrey, Richard C. (1989). Computability and Logic (3rd ed.). Cambridge University Press. pp. 155–156. ISBN 0-521-38026-X. https://archive.org/details/computabilitylog0000bool_c7g3/page/155.
3. ↑ Wolk, Elliot S. (1983), "On the principle of dependent choices and some forms of Zorn's lemma", Canadian Mathematical Bulletin 26 (3): 365–367, doi:10.4153/CMB-1983-062-5
4. ↑ Bernays proved that the axiom of dependent choice implies the axiom of countable choice See esp. p. 86 in Bernays, Paul (1942). "Part III. Infinity and enumerability. Analysis.". Journal of Symbolic Logic. A system of axiomatic set theory 7 (2): 65–89. doi:10.2307/2266303. http://doc.rero.ch/record/290936/files/S0022481200064185.pdf.
5. ↑ For a proof that the Axiom of Countable Choice does not imply the Axiom of Dependent Choice see Jech, Thomas (1973), The Axiom of Choice, North Holland, pp. 130–131, ISBN 978-0-486-46624-8

* Jech, Thomas (2003). Set Theory (Third Millennium ed.). Springer-Verlag. ISBN 3-540-44085-2. OCLC 174929965.

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