Inmathematics, theaxiom of dependent choice, denoted by${\displaystyle \mathsf{D}\mathsf{C}}$, is a weak form of theaxiom of choice (${\displaystyle \mathsf{A}\mathsf{C}}$) that is still sufficient to develop much ofreal analysis. It was introduced byPaul Bernays in a 1942 article inreverse mathematics that explores whichset-theoreticaxioms are needed to develop analysis.^[a]

Ahomogeneous relation${\displaystyle R}$ on${\displaystyle X}$ is called atotal relation if for every${\displaystyle a\in X,}$ there exists some${\displaystyle b\in X}$ such that${\displaystyle aRb}$ is true.

The axiom of dependent choice can be stated as follows: For every nonemptyset${\displaystyle X}$ and every total relation${\displaystyle R}$ on${\displaystyle X,}$ there exists asequence${\displaystyle (x_n{)}_{n\in \mathbb{N}}}$ in${\displaystyle X}$ such that

${\displaystyle x_{n}R{x}_{n+1}}$ for all${\displaystyle n\in \mathbb{N}.}$

In fact,x₀ may be taken to be any desired element ofX. (To see this, apply the axiom as stated above to the set of finite sequences that start withx₀ and in which subsequent terms are in relation${\displaystyle R}$, 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${\displaystyle X}$ above is restricted to be the set of allreal numbers, then the resulting axiom is denoted by${\displaystyle {\mathsf{D}\mathsf{C}}_{\mathbb{R}}.}$

Even without such an axiom, for any${\displaystyle n}$, one can use ordinary mathematical induction to form the first${\displaystyle n}$ terms of such a sequence.The axiom of dependent choice says that we can form a whole (countably infinite) sequence this way.

The axiom${\displaystyle \mathsf{D}\mathsf{C}}$ is the fragment of${\displaystyle \mathsf{A}\mathsf{C}}$ that is required to show the existence of a sequence constructed bytransfinite recursion ofcountable 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.

Over${\displaystyle \mathsf{Z}\mathsf{F}}$ (Zermelo–Fraenkel set theory without the axiom of choice),${\displaystyle \mathsf{D}\mathsf{C}}$ is equivalent to theBaire category theorem for complete metric spaces.^[1]

It is also equivalent over${\displaystyle \mathsf{Z}\mathsf{F}}$ to thedownward Löwenheim–Skolem theorem.^[b][2]

${\displaystyle \mathsf{D}\mathsf{C}}$ is also equivalent over${\displaystyle \mathsf{Z}\mathsf{F}}$ to the statement that everypruned tree with${\displaystyle \omega }$ levels has abranch (proof below).

Furthermore,${\displaystyle \mathsf{D}\mathsf{C}}$ is equivalent to a weakened form ofZorn's lemma; specifically${\displaystyle \mathsf{D}\mathsf{C}}$ is equivalent to the statement that anypartial order such that everywell-orderedchain is finite and bounded, must have a maximal element.^[3]

Proof that${\displaystyle \mathsf{D}\mathsf{C}⟺}$ Every pruned tree with ω levels has a branch

${\displaystyle (⟸)}$ Let${\displaystyle R}$ be an entire binary relation on${\displaystyle X}$. The strategy is to define a tree${\displaystyle T}$ on${\displaystyle X}$ of finite sequences whose neighboring elements satisfy${\displaystyle R.}$ Then a branch through${\displaystyle T}$ is an infinite sequence whose neighboring elements satisfy${\displaystyle R.}$ Start by defining${\displaystyle ⟨x_0,\dots ,x_n⟩\in T}$ if${\displaystyle x_{k}R{x}_{k+1}}$ for Since${\displaystyle R}$ is entire,${\displaystyle T}$ is a pruned tree with${\displaystyle \omega }$ levels. Thus,${\displaystyle T}$ has a branch${\displaystyle ⟨x_0,\dots ,x_n,\dots ⟩.}$ So, for all${\displaystyle n\ge 0:}$${\displaystyle ⟨x_0,\dots ,x_n,x_{n+1}⟩\in T,}$ which implies${\displaystyle x_{n}R{x}_{n+1}.}$ Therefore,${\displaystyle \mathsf{D}\mathsf{C}}$ is true. ${\displaystyle (⟹)}$ Let${\displaystyle T}$ be a pruned tree on${\displaystyle X}$ with${\displaystyle \omega }$ levels. The strategy is to define a binary relation${\displaystyle R}$ on${\displaystyle T}$ so that${\displaystyle \mathsf{D}\mathsf{C}}$ produces a sequence${\displaystyle t_n=⟨x_0,\dots ,x_{f(n)}⟩}$ where${\displaystyle t_{n}R{t}_{n+1}}$ and${\displaystyle f(n)}$ is astrictly increasing function. Then the infinite sequence${\displaystyle ⟨x_0,\dots ,x_k,\dots ⟩}$ is a branch. (This proof only needs to prove this for${\displaystyle f(n)=m+n.}$) Start by defining${\displaystyle uRv}$ if${\displaystyle u}$ is an initial subsequence of${\displaystyle v,\mathrm{length}(u)>0,}$ and${\displaystyle \mathrm{length}(v)=}$${\displaystyle \mathrm{length}(u)+1.}$ Since${\displaystyle T}$ is a pruned tree with${\displaystyle \omega }$ levels,${\displaystyle R}$ is entire. Therefore,${\displaystyle \mathsf{D}\mathsf{C}}$ implies that there is an infinite sequence${\displaystyle t_n}$ such that${\displaystyle t_{n}R{t}_{n+1}.}$ Now${\displaystyle t_0=⟨x_0,\dots ,x_m⟩}$ for some${\displaystyle m\ge 0.}$ Let${\displaystyle x_{m+n}}$ be the last element of${\displaystyle t_n.}$ Then${\displaystyle t_n=⟨x_0,\dots ,x_m,\dots ,x_{m+n}⟩.}$ For all${\displaystyle k\ge 0,}$ the sequence${\displaystyle ⟨x_0,\dots ,x_k⟩}$ belongs to${\displaystyle T}$ because it is an initial subsequence of${\displaystyle t_0(k\le m)}$ or it is a${\displaystyle t_n(k\ge m).}$ Therefore,${\displaystyle ⟨x_0,\dots ,x_k,\dots ⟩}$ is a branch.

Unlike full${\displaystyle \mathsf{A}\mathsf{C}}$,${\displaystyle \mathsf{D}\mathsf{C}}$ is insufficient to prove (given${\displaystyle \mathsf{Z}\mathsf{F}}$) that there is anon-measurable set of real numbers, or that there is a set of real numbers without theproperty of Baire or without theperfect set property. This follows because theSolovay model satisfies${\displaystyle \mathsf{Z}\mathsf{F}+\mathsf{D}\mathsf{C}}$, and every set of real numbers in this model isLebesgue measurable, has the Baire property and has the perfect set property.

The axiom of dependent choice implies theaxiom 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.

• Jech, Thomas (2003).Set Theory (Third Millennium ed.).Springer-Verlag.ISBN 3-540-44085-2.OCLC 174929965.Zbl 1007.03002.

• Axiom of choice

• Articles with short description
• Short description matches Wikidata