nLab dependent choice

Statement

 In set theory

In set theory, the axiom of dependent choice (DC) states that if $X$ is a nonempty (inhabited) set and if $R$ is a total binary relation on $X$, then there exists a sequence $x\colon \mathbb{N} \to X$ such that $(x_n, x_{n+1})$ belongs to $R$ for all $n \in \mathbb{N}$. It is strictly weaker than the full axiom of choice, but strictly stronger than the axiom of countable choice. It is called dependent choice because the available choices for $x_{n+1}$ depend on the choice of $x_n$ made at the previous stage.

There is another way to express the axiom of dependent choice which uses surjections instead of entire relations:

The axiom of dependent choice states that given a family of sets $X_n$ and a family of surjections $f_n:X_{n + 1} \to X_n$ indexed by natural numbers $n \in \mathbb{N}$:

$$X_0 \overset{f_0}\leftarrow X_1 \overset{f_1}\leftarrow X_1 \overset{f_2}\leftarrow \ldots$$

the projection function to $X_0$ from the sequential limit $\underset{\leftarrow}\lim X_n$ of the above diagram is a surjection.

In a topos or pretopos

More generally, given an elementary topos or a pretopos $\mathcal{E}$ with sequential limits, the axiom of dependent choice states that given a sequence of objects $X:\mathbb{N} \to \mathrm{Ob}(\mathcal{E})$ and a dependent sequence of epimorphisms $f_n:\mathrm{hom}_\mathcal{E}(X_{n + 1}, X_{n})$ indexed by $n \in \mathbb{N}$, the projection function to $X_0$ from the sequential limit $\underset{\leftarrow}\lim X_n$ of the above diagram is an epimorphism.

 In dependent type theory

In dependent type theory, the axiom of dependent choice states that if $A$ is an inhabited set and if $R$ is a entire relation on $A$, then then there exists a sequence $a:\mathbb{N} \to A$ such that for all $n:\mathbb{N}$, $R(a(n), a(n + 1))$ holds.

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A, y:A \vdash R(x, y) \; \mathrm{type}}{\Gamma \vdash \mathrm{DC}_{A}^{R}:(\mathrm{isSet}(A) \times [A]) \to \left(\prod_{x:A} \left(\prod_{y:A} \mathrm{isProp}(R(x, y))\right) \times \exists y:A.R(x, y)\right) \to \exists a:\mathbb{N} \to A.\prod_{n:\mathbb{N}} R(a(n), a(n + 1))}$$

If the dependent type theory has a type of all propositions $\mathrm{Prop}$, this can be simplified down to the inference rule

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{DC}_{A}:(\mathrm{isSet}(A) \times [A]) \to \prod_{R:A \times A \to \mathrm{Prop}} \left(\prod_{x:A} \exists y:A.R(x, y)\right) \to \exists a:\mathbb{N} \to A.\prod_{n:\mathbb{N}} R(a(n), a(n + 1))}$$

Applications

Typical applications of dependent choice include Kőnig's lemma? and the Baire category theorem for complete metric spaces.

Acceptability

The axiom of dependent choice holds in the topos of condensed sets and the topos of light condensed sets, assuming the axiom of choice in the category of sets.

For a number of schools of constructive mathematics, dependent choice is considered an acceptable alternative to full AC, and the principle holds (assuming that it holds in Set) in quite a few toposes of interest in which full AC doesn’t hold, including all presheaf toposes and the effective topos. In fact, it follows from the stronger presentation axiom, which holds in many of these toposes and also has a constructive justification.

Like countable choice, it fails for sheaves over the space of real numbers.

Variations

Dependent $n$-choice and $\infty$-choice

In dependent type theory, the axiom of dependent choice requires that $A$ be a set. This can be generalised to the axiom of dependent $n$-choice by replacing the set requirement with an $n$-truncation requirement for $A$: if $A$ is an inhabited $n$-truncated type and if $R$ is a entire relation on $A$, then then there exists a sequence $a:\mathbb{N} \to A$ such that for all $n:\mathbb{N}$, $R(a(n), a(n + 1))$ holds.

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A, y:A \vdash R(x, y) \; \mathrm{type}}{\Gamma \vdash \mathrm{DC}_{n, A}^{R}:(\mathrm{is}n\mathrm{type}(A) \times [A]) \to \left(\prod_{x:A} \left(\prod_{y:A} \mathrm{isProp}(R(x, y))\right) \times \exists y:A.R(x, y)\right) \to \exists a:\mathbb{N} \to A.\prod_{n:\mathbb{N}} R(a(n), a(n + 1))}$$

These are stronger axioms as $n$ increases.

In the limit as $n$ goes to infinity, this is called the axiom of dependent infinity-choice, where $A$ doesn’t have any truncation requirements: if $A$ is an inhabited type and if $R$ is a entire relation on $A$, then then there exists a sequence $a:\mathbb{N} \to A$ such that for all $n:\mathbb{N}$, $R(a(n), a(n + 1))$ holds.

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A, y:A \vdash R(x, y) \; \mathrm{type}}{\Gamma \vdash \mathrm{DC}_{\infty, A}^{R}:[A] \to \left(\prod_{x:A} \left(\prod_{y:A} \mathrm{isProp}(R(x, y))\right) \times \exists y:A.R(x, y)\right) \to \exists a:\mathbb{N} \to A.\prod_{n:\mathbb{N}} R(a(n), a(n + 1))}$$

Dependent choice for the natural numbers

Dependent choice for the natural numbers $\mathrm{DC}_\mathbb{N}$ states that if $R$ is a entire relation on $\mathbb{N}$, then there exists a sequence $x$ in $\mathbb{N}$ such that $R(x_n, x_{n+1})$ holds for all $n \in \mathbb{N}$. $\mathrm{DC}_\mathbb{N}$ is enough to prove that every Dedekind real number is a Cauchy real number (the converse is always true).

In dependent type theory this is given by the following inference rule:

$$\frac{\Gamma, x:\mathbb{N}, y:\mathbb{N} \vdash R(x, y) \; \mathrm{type}}{\Gamma \vdash \mathrm{DC}_\mathbb{N}^{R}:\left(\prod_{x:\mathbb{N}} \left(\prod_{y:\mathbb{N}} \mathrm{isProp}(R(x, y))\right) \times \exists y:\mathbb{N}.R(x, y)\right) \to \exists a:\mathbb{N} \to \mathbb{N}.\prod_{n:\mathbb{N}} R(a(n), a(n + 1))}$$

If the dependent type theory has a type of all propositions $\mathrm{Prop}$, this can be simplified down to the axiom

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{DC}_\mathbb{N}:\prod_{R:\mathbb{N} \times \mathbb{N} \to \mathrm{Prop}} \left(\prod_{x:\mathbb{N}} \exists y:\mathbb{N}.R(x, y)\right) \to \exists a:\mathbb{N} \to \mathbb{N}.\prod_{n:\mathbb{N}} R(a(n), a(n + 1))}$$

Related concepts

• axiom of choice

• axiom of countable choice

References

• Wikipedia, Axiom of dependent choice

• Felix Cherubini, Thierry Coquand, Freek Geerligs, Hugo Moeneclaey, A Foundation for Synthetic Stone Duality (arXiv:2412.03203, summary)

• Mathieu Anel, Reid Barton, Choice axioms and Postnikov completeness (arXiv:2403.19772)

category: foundational axiom