nLab countable choice

Idea

The axiom of countable choice ($CC$), also called $AC_\omega$ or $AC_N$, is a weak form of the axiom of choice; it says that the set of natural numbers is a projective object in Set. (Recall that the full axiom of choice states that every set is projective.)

In classical mathematics, countable choice is usually accepted because the full axiom of choice is accepted. In constructive mathematics the situation is more subtle. For varying reasons, some schools of constructive mathematics accept countable choice (though they reject the full axiom of choice). On the other hand, countable choice is not valid in the internal logic of a general topos, so if one desires this level of generality then it should not be assumed. There are also philosophical constructivist arguments against it. Fred Richman (RichmanFTA) has said that

Countable choice is a blind spot for constructive mathematicians in much the same way as excluded middle is for classical mathematicians.

All the reasoning in this page is constructive.

Definition

In set theory

More explicitly, let $X$ be any set and let $p\colon X \to \mathbf{N}$ be a surjection. Then the axiom of countable choice states that $p$ has a section. If you phrase the axiom of choice in terms of entire relations, then countable choice states that any entire relation from $\mathbf{N}$ to any other set contains (in the 2-poset Rel) a functional entire relation.

In dependent type theory

In dependent type theory, countable choice says that the dependent product of a family of inhabited sets indexed by the natural numbers is itself inhabited:

$$\frac{\Gamma, n:\mathbb{N} \vdash A(n) \; \mathrm{type}}{\Gamma \vdash \mathrm{AC}_\mathbb{N}^A:\left(\prod_{n:\mathbb{N}} \mathrm{isSet}(A(n))\right) \to \left(\prod_{n:\mathbb{N}} [A(n)]\right) \to \left[\prod_{n:\mathbb{N}} A(n)\right]}$$

Alternatively, the axiom of countable choice states that given a family of sets $A(n)$ indexed by natural numbers $n$ and a family of propositions $P(n, x)$ indexed by natural numbers $n$ and element $x:A(n)$, if for all natural numbers $n$ there merely exists an element $x:A(n)$ such that $P(n, x)$, then there merely exists a dependent function $g:\prod_{n:\mathbb{N}} A(n)$ such that for all natural numbers $n$, $P(n, g(n))$:

$$\frac{\Gamma, n:\mathbb{N} \vdash A(n) \; \mathrm{type} \quad \Gamma, n:\mathbb{N}, x:A(n) \vdash P(n, x) \; \mathrm{type}}{\Gamma \vdash \mathrm{AC}_\mathbb{N}^{A, P}:\left(\prod_{n:\mathbb{N}} \mathrm{isSet}(A(n)) \times \prod_{x:A(n)} \mathrm{isProp}(P(n, x))\right) \to \left(\prod_{n:\mathbb{N}} \exists x:A(n).P(n, x)\right) \to \exists g:\prod_{n:\mathbb{N}} A(n).\prod_{n:\mathbb{N}} P(n, g(n))}$$

One can also use the version using surjections: let $A$ be a set. Then the axiom of countable choice states that for all functions $f:A \to \mathbb{N}$, if for all natural numbers $n:\mathbb{N}$ there merely exists an element $x:A$ such that $f(x) = n$, then there merely exists a function $g:\mathbb{N} \to A$ such that $f(g(n)) = n$ for all $n:\mathbb{N}$.

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{AC}_\mathbb{N}^{A}:\mathrm{isSet}(A) \times \left(\prod_{f:A \to \mathbb{N}} \left(\prod_{n:\mathbb{N}} \exists x:A.f(x) = n\right) \to \exists g:\mathbb{N} \to A.\prod_{n:\mathbb{N}} f(g(n)) = n\right)}$$

The version using surjections has the advantage that if the dependent type theory has type variables and impredicative polymorphism, countable choice can be expressed as an actual axiom rather than an axiom schema:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{AC}_\mathbb{N}:\Pi A.\mathrm{isSet}(A) \times \left(\prod_{f:A \to \mathbb{N}} \left(\prod_{n:\mathbb{N}} \exists x:A.f(x) = n\right) \to \exists g:\mathbb{N} \to A.\prod_{n:\mathbb{N}} f(g(n)) = n\right)}$$

Consequences

Here we collect some consequences of the countable axiom of choice.

• The Cauchy real numbers and Dedekind real numbers coincide (the law of excluded middle also separately implies this).

• The Rosolini dominance is a dominance (the law of excluded middle also separately implies this).

• countable unions of countable sets are countable

Variations

$COSHEP$ & $DC$

Unlike the full axiom of choice, countable choice is often considered to be a constructively acceptable principle. In particular, it does not imply the principle of excluded middle. It is a consequence of COSHEP. A stronger version of countable choice, also a consequence of $COSHEP$, is the axiom of dependent choice ($DC$). In general, $DC$ is enough to justify results in analysis involving sequences.

$\mathrm{AC}_{00}$

Sometimes in foundations it is useful to consider a weaker version of countable choice, called $\mathrm{AC}_{00}$ or $\mathrm{AC}_{\mathbb{N},\mathbb{N}}$. This states that any entire relation $R$ from $\mathbb{N}$ to itself contains a functional entire relation, i.e. there exists a sequence $x$ in $\mathbb{N}$ such that $R(n, x_n)$ holds. In terms of surjections, the axiom states that any surjection $p\colon X \to \mathbb{N}$ has a section if $X$ is a subset of $\mathbb{N} \times \mathbb{N}$ and $p$ is the restriction to $X$ of a product projection. $AC_{00}$ is enough to prove that every Dedekind real number is a Cauchy real number (the converse is always true).

Weak countable choice

The axiom of weak countable choice ($WCC$) states that a surjection $p\colon X \to \mathbf{N}$ has a section if, whenever $m \neq n$, at least one of the preimages $p^*(m)$ and $p^*(n)$ is a singleton. $WCC$ follows (for different reasons) from either $CC$ or excluded middle. On the other hand, $WCC$ is enough to prove that the fundamental theorem of algebra in the sense that every non-constant complex polynomial has a root; see Bridges et al (1998). It is an open question whether $WCC$ implies that the Dedekind reals and Cauchy reals coincide, see King et al (2024).

Another weak countable choice

An axiom variously called $AC_{weak}$ and $\mathrm{AC}_{\mathrm{N},2}$ is countable choice for subsets of $\{0,1\}$. This states that any entire relation $R$ from $\mathbb{N}$ to $\{0,1\}$ contains a functional entire relation, i.e. there exists a sequence $x$ in $\{0,1\}$ such that $R(n, x_n)$ holds. Like $WCC$ above, this also follows from either $CC$ or excluded middle. It is enough to prove the equivalence of the Dedekind reals and Cauchy reals. See Saving et al (2021), King et al (2024).

Very weak countable choice

An even weaker form of countable choice was proposed by Martin Escardo; it states that any surjection of the form $A \sqcup (\mathbf{N}\times B) \to \mathbf{N}$ has a section, where $A\to \mathbf{N}$ is a decidable subset and $B$ is an arbitrary set with $\mathbf{N}\times B \to \mathbf{N}$ the projection. This follows from WCC and also from the limited principle of omniscience; see the constructivenews discussion.

Using a dominance

In constructive mathematics, one can express weaker versions of the axiom of countable choice by using an arbitrary dominance $\Sigma \subseteq \Omega$ and restricting countable choice to the $\Sigma$-open relations: For any set $A$, any entire open relation $R:\mathbb{N} \times A \to \Sigma$ from $\mathbb{N}$ to $A$ contains a functional entire relation, i.e. there exists a sequence $x$ in $A$ such that $R(n, x_n) = \top$.

Similarly, we can express weaker versions of $\mathrm{AC}_{\mathbb{N},\mathbb{N}}$ and $\mathrm{AC}_{\mathbb{N},2}$ using a dominance $\Sigma$:

• $\mathrm{AC}_{\mathbb{N},\mathbb{N},\Sigma}$ states that any entire open relation $R:\mathbb{N} \times \mathbb{N} \to \Sigma$ from $\mathbb{N}$ to itself contains a functional entire relation, i.e. there exists a sequence $x$ in $\mathbb{N}$ such that $R(n, x_n) = \top$.

• $\mathrm{AC}_{\mathbb{N},2,\Sigma}$ states that any entire open relation $R:\mathbb{N} \times \{0, 1\} \to \Sigma$ from $\mathbb{N}$ to the boolean domain $\{0, 1\}$ contains a functional entire relation, i.e. there exists a sequence $x$ in $\{0, 1\}$ such that $R(n, x_n) = \top$.

Theorem

For $\Sigma$ an $\mathbb{N}$-overt dominance, any of the above axioms are sufficient to prove that the Cauchy real numbers $\mathbb{R}_C$ and the Dedekind real numbers $\mathbb{R}_\Sigma$ constructed out of Dedekind cuts valued in $\Sigma$ are equivalent.

Proof

It suffices to prove the theorem using $\mathrm{AC}_{\mathbb{N},2,\Sigma}$, since both $\mathrm{AC}_{\mathbb{N},\mathbb{N},\Sigma}$ and the more general $\Sigma$-open countable choice both imply $\mathrm{AC}_{\mathbb{N},2,\Sigma}$.

Adopting the proof of Corollary 11.4.3 from the HoTT book, suppose that $\mathrm{AC}_{\mathbb{N},2,\Sigma}$ holds. The set

$$S = \{ (q, r) \in \mathbb{Q} \times \mathbb{Q} \vert q \lt r \}$$

is equivalent to $\mathbb{N}$, so we may apply $\mathrm{AC}_{\mathbb{N},2,\Sigma}$ to the statement that $x$ is located:

$$\forall q,r \in S.(q \lt x) \vee (x \lt r)$$

Note that $(q \lt x) \vee (x \lt r)$ is expressible as an existential statement

$$\exists b \in \{0, 1\}.(b = 0) \Rightarrow (q \lt x) \times (b = 1) \Rightarrow (x \lt r)$$

and $(b = 0) \Rightarrow (q \lt x) \times (b = 1) \Rightarrow (x \lt r)$ is $\Sigma$-open, since the order relation $x \lt y$ on $\mathbb{R}_\Sigma$ is $\Sigma$-open by theorem 11.2.14 from the HoTT book and equality on $\{0, 1\}$ is a decidable relation, and given a decidable proposition $P$ (which implies that both $P$ and $\neg P$ are $\Sigma$-open) and a $\Sigma$-open proposition $Q$, $P \Rightarrow Q$ is logically equivalent to $(P \wedge Q) \vee \neg P$ and thus $\Sigma$-open.

The (curried form) of the choice function from $\mathrm{AC}_{\mathbb{N},2,\Sigma}$ is then the locator

$$\prod_{q, r \in \mathbb{Q}} (q \lt r) \to (q \lt x) + (x \lt r)$$

and lemma 11.4.1 from the HoTT book implies that $\mathbb{R}_C$ and $\mathbb{R}_\Sigma$ coincide.

Topos violating the CAC

One easy example is the category $Sh([0,1])$, the sheaves on the unit interval, since in that topos the Dedekind real numbers and the Cauchy real numbers are not isomorphic, and this is a consequence of the internal countable choice.

Discussion here.

In higher category theory

To formulate a version of the axiom of countable choice in a higher category, one has to make an appropriate choice of the meaning of “epimorphism”. In most cases, it is best to choose effective epimorphism in an (infinity,1)-category or a related notion such as eso morphisms.

There are multiple version of the axiom of countable choice, depending on what truncation requirements, if any, are applied to the domain.

• An $(\infty,1)$-category satisfies the axiom of countable $n$-choice, or $CC_n$, if every $n$-truncated morphism? into the natural numbers object has a section. We write $CC_\infty$ for the axiom of countable infinity-choice: the statement that every morphism with codomain the natural numbers object has a section.

These are stronger axioms as $n$ increases.

There are also “internal” versions of these axioms.

• In homotopy type theory (the internal logic of an $(\infty,1)$-topos), the internal version of $CC_n$ is “every surjection onto the natural numbers type with $n$-type fibers has a section”, or equivalently

  $$\prod_{(Y:\mathbb{N}\to n Type)} \Big( \prod_{(x:\mathbb{N})} \Vert Y(x)\Vert \to \Big\Vert \prod_{(x:\mathbb{N})} Y(x) \Big\Vert \Big)$$

• More generally, we can replace the $(-1)$-truncation by the $k$-truncation to obtain a family of axioms $CC_{k,n}$.

• We can also replace the $(-1)$-truncation by the assertion of $k$-connectedness, obtaining the axiom of countable $k$-connected choice.

References

• Fred Richman, Douglas Bridges, Peter Schuster, A weak countable choice principle. Proceedings of the American Mathematical Society 128(9):2749-2752, March 2000. [doi:10.1090/S0002-9939-00-05327-2]

• Martin Escardo et. al., Special case of countable choice, message and discussion to the constructivenews list, google groups

• Fred Richman, The fundamental theorem of algebra: a constructive development without choice. Pacific Journal of Mathematics 196 1 (2000) 213–230 [doi:10.2140/pjm.2000.196.213, pdf]

• Mark Saving et al (2021). A weak form of countable choice. MathOverflow.

• Christopher King et al (2024). Does weak countable choice imply that the Cauchy reals are Dedekind complete? MathOverflow.

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

• Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)

See also

• Wikipedia, Axiom of countable choice

category: foundational axiom