Axiom of choice

In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to construct a new set by choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family ${\displaystyle (S_{i})_{i\in I}}$ of

${\displaystyle (x_{i})_{i\in I}}$

${\displaystyle x_{i}\in S_{i}}$

${\displaystyle i\in I}$

In many cases, a set created by choosing elements can be made without invoking the axiom of choice, particularly if the number of sets from which to choose the elements is finite, or if a canonical rule on how to choose the elements is available — some distinguishing property that happens to hold for exactly one element in each set. An illustrative example is sets picked from the natural numbers. From such sets, one may always select the smallest number, e.g. given the sets {{4, 5, 6}, {10, 12}, {1, 400, 617, 8000}}, the set containing each smallest element is {4, 10, 1}. In this case, "select the smallest number" is a choice function. Even if infinitely many sets are collected from the natural numbers, it will always be possible to choose the smallest element from each set to produce a set. That is, the choice function provides the set of chosen elements. But no definite choice function is known for the collection of all non-empty subsets of the real numbers. In that case, the axiom of choice must be invoked.

Bertrand Russell coined an analogy: for any (even infinite) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate collection (i.e. set) of shoes; this makes it possible to define a choice function directly. For an infinite collection of pairs of socks (assumed to have no distinguishing features such as being a left sock rather than a right sock), there is no obvious way to make a function that forms a set out of selecting one sock from each pair without invoking the axiom of choice.^[3]

Although originally controversial, the axiom of choice is now used without reservation by most mathematicians,

A choice function (also called selector or selection) is a function f, defined on a collection X of nonempty sets, such that for every set A in X, f(A) is an element of A. With this concept, the axiom can be stated:

Formally, this may be expressed as follows:

${\displaystyle \forall X\left[\varnothing \notin X\implies \exists f\colon X\rightarrow \bigcup _{A\in X}A\quad \forall A\in X\,(f(A)\in A)\right]\,.}$

Thus, the negation of the axiom may be expressed as the existence of a collection of nonempty sets which has no choice function. Formally, this may be derived making use of the logical equivalence of ${\displaystyle \neg \forall X\left[P(X)\to Q(X)\right]}$ to ${\displaystyle \exists X\left[P(X)\land \neg Q(X)\right]}$.

Each choice function on a collection X of nonempty sets is an element of the Cartesian product of the sets in X. This is not the most general situation of a Cartesian product of a family of sets, where a given set can occur more than once as a factor; however, one can focus on elements of such a product that select the same element every time a given set appears as factor, and such elements correspond to an element of the Cartesian product of all distinct sets in the family. The axiom of choice asserts the existence of such elements; it is therefore equivalent to:

Given any family of nonempty sets, their Cartesian product is a nonempty set.

In this article and other discussions of the Axiom of Choice the following abbreviations are common:

• AC – the Axiom of Choice. More rarely, AoC is used.^[5]
• ZF – Zermelo–Fraenkel set theory omitting the Axiom of Choice.
• ZFC – Zermelo–Fraenkel set theory, extended to include the Axiom of Choice.

There are many other equivalent statements of the axiom of choice. These are equivalent in the sense that, in the presence of other basic axioms of set theory, they imply the axiom of choice and are implied by it.

One variation avoids the use of choice functions by, in effect, replacing each choice function with its range:

Given any set X, if the empty set is not an element of X and the elements of X are

pairwise disjoint, then there exists a set C such that its intersection with any of the elements of X contains exactly one element.^[6]

This guarantees for any partition of a set X the existence of a subset C of X containing exactly one element from each part of the partition.

Another equivalent axiom only considers collections X that are essentially powersets of other sets:

For any set A, the power set of A (with the empty set removed) has a choice function.

Authors who use this formulation often speak of the choice function on A, but this is a slightly different notion of choice function. Its domain is the power set of A (with the empty set removed), and so makes sense for any set A, whereas with the definition used elsewhere in this article, the domain of a choice function on a collection of sets is that collection, and so only makes sense for sets of sets. With this alternate notion of choice function, the axiom of choice can be compactly stated as

Every set has a choice function.^[7]

which is equivalent to

For any set A there is a function f such that for any non-empty subset B of A, f(B) lies in B.

The negation of the axiom can thus be expressed as:

There is a set A such that for all functions f (on the set of non-empty subsets of A), there is a B such that f(B) does not lie in B.

The usual statement of the axiom of choice does not specify whether the collection of nonempty sets is finite or infinite, and thus implies that every finite collection of nonempty sets has a choice function. However, that particular case is a theorem of the Zermelo–Fraenkel set theory without the axiom of choice (ZF); it is easily proved by the principle of finite induction.^[8] In the even simpler case of a collection of one set, a choice function just corresponds to an element, so this instance of the axiom of choice says that every nonempty set has an element; this holds trivially. The axiom of choice can be seen as asserting the generalization of this property, already evident for finite collections, to arbitrary collections.

Until the late 19th century, the axiom of choice was often used implicitly, although it had not yet been formally stated. For example, after having established that the set X contains only non-empty sets, a mathematician might have said "let F(s) be one of the members of s for all s in X" to define a function F. In general, it is impossible to prove that F exists without the axiom of choice, but this seems to have gone unnoticed until

The nature of the individual nonempty sets in the collection may make it possible to avoid the axiom of choice even for certain infinite collections. For example, suppose that each member of the collection X is a nonempty subset of the natural numbers. Every such subset has a smallest element, so to specify our choice function we can simply say that it maps each set to the least element of that set. This gives us a definite choice of an element from each set, and makes it unnecessary to add the axiom of choice to our axioms of set theory.

The difficulty appears when there is no natural choice of elements from each set. If we cannot make explicit choices, how do we know that our selection forms a legitimate set (as defined by the other ZF axioms of set theory)? For example, suppose that X is the set of all non-empty subsets of the real numbers. First we might try to proceed as if X were finite. If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently we shall never be able to produce a choice function for all of X. Next we might try specifying the least element from each set. But some subsets of the real numbers do not have least elements. For example, the open interval (0,1) does not have a least element: if x is in (0,1), then so is x/2, and x/2 is always strictly smaller than x. So this attempt also fails.

Additionally, consider for instance the unit circle S, and the action on S by a group G consisting of all rational rotations, that is, rotations by angles which are rational multiples of π. Here G is countable while S is uncountable. Hence S breaks up into uncountably many orbits under G. Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset X of S with the property that all of its translates by G are disjoint from X. The set of those translates partitions the circle into a countable collection of pairwise disjoint sets, which are all pairwise congruent. Since X is not measurable for any rotation-invariant countably additive finite measure on S, finding an algorithm to form a set from selecting a point in each orbit requires that one add the axiom of choice to our axioms of set theory. See non-measurable set for more details.

In classical arithmetic, the natural numbers are well-ordered: for every nonempty subset of the natural numbers, there is a unique least element under the natural ordering. In this way, one may specify a set from any given subset. One might say, "Even though the usual ordering of the real numbers does not work, it may be possible to find a different ordering of the real numbers which is a well-ordering. Then our choice function can choose the least element of every set under our unusual ordering." The problem then becomes that of constructing a well-ordering, which turns out to require the axiom of choice for its existence; every set can be well-ordered if and only if the axiom of choice holds.

A proof requiring the axiom of choice may establish the existence of an object without explicitly

The axiom of choice proves the existence of these intangibles (objects that are proved to exist, but which cannot be explicitly constructed), which may conflict with some philosophical principles.[10] Because there is no canonical well-ordering of all sets, a construction that relies on a well-ordering may not produce a canonical result, even if a canonical result is desired (as is often the case in category theory). This has been used as an argument against the use of the axiom of choice.

Another argument against the axiom of choice is that it implies the existence of objects that may seem counterintuitive.^[11] One example is the Banach–Tarski paradox, which says that it is possible to decompose the 3-dimensional solid unit ball into finitely many pieces and, using only rotations and translations, reassemble the pieces into two solid balls each with the same volume as the original. The pieces in this decomposition, constructed using the axiom of choice, are non-measurable sets.

Moreover, paradoxical consequences of the axiom of choice for the

Despite these seemingly paradoxical results, most mathematicians accept the axiom of choice as a valid principle for proving new results in mathematics. But the debate is interesting enough that it is considered notable when a theorem in ZFC (ZF plus AC) is logically equivalent (with just the ZF axioms) to the axiom of choice, and mathematicians look for results that require the axiom of choice to be false, though this type of deduction is less common than the type that requires the axiom of choice to be true.

Theorems of ZF hold true in any model of that theory, regardless of the truth or falsity of the axiom of choice in that particular model. The implications of choice below, including weaker versions of the axiom itself, are listed because they are not theorems of ZF. The Banach–Tarski paradox, for example, is neither provable nor disprovable from ZF alone: it is impossible to construct the required decomposition of the unit ball in ZF, but also impossible to prove there is no such decomposition. Such statements can be rephrased as conditional statements—for example, "If AC holds, then the decomposition in the Banach–Tarski paradox exists." Such conditional statements are provable in ZF when the original statements are provable from ZF and the axiom of choice.

As discussed above, in the classical theory of ZFC, the axiom of choice enables

Different choice principles have been thoroughly studied in the constructive contexts and the principles' status varies between different school and varieties of the constructive mathematics. Some results in constructive set theory use the axiom of countable choice or the axiom of dependent choice, which do not imply the law of the excluded middle. Errett Bishop, who is notable for developing a framework for constructive analysis, argued that an axiom of choice was constructively acceptable, saying

A choice function exists in constructive mathematics, because a choice is implied by the very meaning of existence.^[15]

Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.^[16]

It has been known since as early as 1922 that the axiom of choice may fail in a variant of ZF with urelements, through the technique of permutation models introduced by Abraham Fraenkel^[17] and developed further by Andrzej Mostowski.^[18] The basic technique can be illustrated as follows: Let x_n and y_n be distinct urelements for n=1, 2, 3..., and build a model where each set is symmetric under the interchange x_n ↔ y_n for all but a finite number of n. Then the set X = {{x₁, y₁}, {x₂, y₂}, {x₃, y₃}, ...} can be in the model but sets such as {x₁, x₂, x₃, ...} cannot, and thus X cannot have a choice function.

In 1938,

Together these results establish that the axiom of choice is logically independent of ZF. The assumption that ZF is consistent is harmless because adding another axiom to an already inconsistent system cannot make the situation worse. Because of independence, the decision whether to use the axiom of choice (or its negation) in a proof cannot be made by appeal to other axioms of set theory. It must be made on other grounds.

One argument in favor of using the axiom of choice is that it is convenient because it allows one to prove some simplifying propositions that otherwise could not be proved. Many theorems provable using choice are of an elegant general character: the cardinalities of any two sets are comparable, every nontrivial

The axiom of choice is not the only significant statement that is independent of ZF. For example, the generalized continuum hypothesis (GCH) is not only independent of ZF, but also independent of ZFC. However, ZF plus GCH implies AC, making GCH a strictly stronger claim than AC, even though they are both independent of ZF.

The axiom of constructibility and the generalized continuum hypothesis each imply the axiom of choice and so are strictly stronger than it. In class theories such as Von Neumann–Bernays–Gödel set theory and Morse–Kelley set theory, there is an axiom called the axiom of global choice that is stronger than the axiom of choice for sets because it also applies to proper classes. The axiom of global choice follows from the axiom of limitation of size. Tarski's axiom, which is used in Tarski–Grothendieck set theory and states (in the vernacular) that every set belongs to some Grothendieck universe, is stronger than the axiom of choice.

There are important statements that, assuming the axioms of ZF but neither AC nor ¬AC, are equivalent to the axiom of choice.^[22] The most important among them are Zorn's lemma and the well-ordering theorem. In fact, Zermelo initially introduced the axiom of choice in order to formalize his proof of the well-ordering theorem.

• Set theory
  • bijective map
    between the sets A and A×A.
  • Trichotomy
    : If two sets are given, then either they have the same cardinality, or one has a smaller cardinality than the other.
  • Given two non-empty sets, one has a surjection to the other.
  • Every surjective function has a right inverse.
  • The Cartesian product of any family of nonempty sets is nonempty. In other words, every family of nonempty sets has a choice function (i.e. a function which maps each of the nonempty sets to one of its elements).
  • König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals. (The reason for the term "colloquially" is that the sum or product of a "sequence" of cardinals cannot itself be defined without some aspect of the axiom of choice.)
  • initial ordinal
    .
  • Zorn's lemma: Every non-empty partially ordered set in which every chain (i.e., totally ordered subset) has an upper bound contains at least one maximal element.
  • Hausdorff maximal principle: Every partially ordered set has a maximal chain. Equivalently, in any partially ordered set, every chain can be extended to a maximal chain.
  • Tukey's lemma: Every non-empty collection of finite character
    has a maximal element with respect to inclusion.
  • Antichain principle: Every partially ordered set has a maximal antichain. Equivalently, in any partially ordered set, every antichain can be extended to a maximal antichain.
  • The powerset of any ordinal can be well-ordered.
• Abstract algebra
  • Every vector space has a basis (i.e., a linearly independent spanning subset). In other words, vector spaces are equivalent to free modules.^[23]
  • Krull's theorem: Every unital ring (other than the trivial ring) contains a maximal ideal. Equivalently, in any nontrivial unital ring, every ideal can be extended to a maximal ideal.
  • For every non-empty set S there is a binary operation defined on S that gives it a group structure.^[24] (A cancellative binary operation is enough, see group structure and the axiom of choice.)
  • Every free abelian group is projective.^[25]
  • Baer's criterion: Every divisible abelian group is injective.^[25]
  • Every set is a projective object in the category Set of sets.^[26][27]
• Functional analysis
  • The closed unit ball of the dual of a normed vector space over the reals has an extreme point.
• Point-set topology
  • The Cartesian product of any family of connected topological spaces is connected.
  • Tychonoff's theorem: The Cartesian product of any family of compact topological spaces is compact.
  • In the product topology, the closure of a product of subsets is equal to the product of the closures.
• Mathematical logic
  • If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem; see the section "Weaker forms" below.
• Graph theory
  • Every
    connected graph has a spanning tree. Equivalently, every nonempty graph has a spanning forest.^[28]

• Every small category has a skeleton.
• If two small categories are weakly equivalent, then they are equivalent.
• Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint (the Freyd adjoint functor theorem).

There are several weaker statements that are not equivalent to the axiom of choice but are closely related. One example is the axiom of dependent choice (DC). A still weaker example is the axiom of countable choice (AC_ω or CC), which states that a choice function exists for any countable set of nonempty sets. These axioms are sufficient for many proofs in elementary mathematical analysis, and are consistent with some principles, such as the Lebesgue measurability of all sets of reals, that are disprovable from the full axiom of choice.

Given an ordinal parameter α ≥ ω+2 — for every set S with rank less than α, S is well-orderable. Given an ordinal parameter α ≥ 1 — for every set S with Hartogs number less than ω_α, S is well-orderable. As the ordinal parameter is increased, these approximate the full axiom of choice more and more closely.

Other choice axioms weaker than axiom of choice include the

One of the most interesting aspects of the axiom of choice is the large number of places in mathematics where it shows up. Here are some statements that require the axiom of choice in the sense that they are not provable from ZF but are provable from ZFC (ZF plus AC). Equivalently, these statements are true in all models of ZFC but false in some models of ZF.

• Set theory
  • The
    ultrafilter lemma
    (with ZF) can be used to prove the Axiom of choice for finite sets: Given ${\displaystyle I\neq \varnothing }$ and a collection ${\displaystyle \left(X_{i}\right)_{i\in I}}$ of non-empty finite sets, their product ${\displaystyle \prod _{i\in I}X_{i}}$ is not empty.^[29]
  • The
    countable sets is countable (this requires countable choice
    but not the full axiom of choice).
  • If the set A is
    Dedekind infinite).^[30]
  • Eight definitions of a finite set are equivalent.^[31]
  • Every infinite game ${\displaystyle G_{S}}$ in which ${\displaystyle S}$ is a Borel subset of Baire space is determined.
• Every infinite cardinal κ satisfies 2×κ = κ.^[32]
• Measure theory
  • The
    Lebesgue measurable
    .
  • There exist Lebesgue-measurable subsets of the real numbers that are not Borel sets. That is, the Borel σ-algebra on the real numbers (which is generated by all real intervals) is distinct from the Lebesgue-measure σ-algebra on the real numbers.
  • The Hausdorff paradox.
  • The Banach–Tarski paradox.
• Algebra
  • Every field has an algebraic closure.
  • Every
    transcendence basis
    .
  • Every infinite-dimensional vector space contains an infinite linearly independent subset (this requires dependent choice, but not the full axiom of choice).
  • Stone's representation theorem for Boolean algebras needs the Boolean prime ideal theorem.
  • The Nielsen–Schreier theorem, that every subgroup of a free group is free.
  • The additive groups of
    R and C are isomorphic.^[33][34]
• Functional analysis
  • The Hahn–Banach theorem in functional analysis, allowing the extension of linear functionals.
  • The theorem that every Hilbert space has an orthonormal basis.
  • The
    compactness
    of sets of functionals.
  • The
    complete metric spaces, and its consequences, such as the open mapping theorem and the closed graph theorem
    .
  • On every infinite-dimensional topological vector space there is a discontinuous linear map.
• General topology
  • A uniform space is compact if and only if it is complete and totally bounded.
  • Every Tychonoff space has a Stone–Čech compactification.
• Mathematical logic
  • Gödel's completeness theorem for first-order logic: every consistent set of first-order sentences has a completion. That is, every consistent set of first-order sentences can be extended to a maximal consistent set.
  • The compactness theorem: If ${\displaystyle \Sigma }$ is a set of
    first-order (or alternatively, zero-order) sentences such that every finite
    subset of ${\displaystyle \Sigma }$ has a
    model
    , then ${\displaystyle \Sigma }$ has a model.[35]

There are several historically important set-theoretic statements implied by AC whose equivalence to AC is open. Zermelo cited the partition principle, which was formulated before AC itself, as a justification for believing AC. In 1906, Russell declared PP to be equivalent, but whether the partition principle implies AC is the oldest open problem in set theory,^[36] and the equivalences of the other statements are similarly hard old open problems. In every known model of ZF where choice fails, these statements fail too, but it is unknown whether they can hold without choice.

• Set theory
  • Partition principle: if there is a surjection from A to B, there is an injection from B to A. Equivalently, every partition P of a set S is less than or equal to S in size.
  • Converse Schröder–Bernstein theorem: if two sets have surjections to each other, they are equinumerous.
  • Weak partition principle: if there is an injection and a surjection from A to B, then A and B are equinumerous. Equivalently, a partition of a set S cannot be strictly larger than S. If WPP holds, this already implies the existence of a non-measurable set. Each of the previous three statements is implied by the preceding one, but it is unknown if any of these implications can be reversed.
  • There is no infinite decreasing sequence of cardinals. The equivalence was conjectured by Schoenflies in 1905.
• Abstract algebra
  • Hahn embedding theorem: Every ordered abelian group G order-embeds as a subgroup of the additive group ${\displaystyle \mathbb {R} ^{\Omega }}$ endowed with a
    lexicographical order
    , where Ω is the set of Archimedean equivalence classes of G. This equivalence was conjectured by Hahn in 1907.

If we abbreviate by BP the claim that every set of real numbers has the property of Baire, then BP is stronger than ¬AC, which asserts the nonexistence of any choice function on perhaps only a single set of nonempty sets. Strengthened negations may be compatible with weakened forms of AC. For example, ZF + DC^[37] + BP is consistent, if ZF is.

It is also consistent with ZF + DC that every set of reals is

Quine's system of axiomatic set theory, New Foundations (NF), takes its name from the title ("New Foundations for Mathematical Logic") of the 1937 article that introduced it. In the NF axiomatic system, the axiom of choice can be disproved.^[38]

There are models of Zermelo-Fraenkel set theory in which the axiom of choice is false. We shall abbreviate "Zermelo-Fraenkel set theory plus the negation of the axiom of choice" by ZF¬C. For certain models of ZF¬C, it is possible to validate the negation of some standard ZFC theorems. As any model of ZF¬C is also a model of ZF, it is the case that for each of the following statements, there exists a model of ZF in which that statement is true.

• The negation of the
  weak partition principle
  : There is a set that can be partitioned into strictly more equivalence classes than the original set has elements, and a function whose domain is strictly smaller than its range. In fact, this is the case in all known models.
• There is a function f from the real numbers to the real numbers such that f is not continuous at a, but f is
  sequentially continuous
  at a, i.e., for any sequence {x_n} converging to a, lim_n f(x_n)=f(a).
• There is an infinite set of real numbers without a countably infinite subset.
• The real numbers are a countable union of countable sets.^[39] This does not imply that the real numbers are countable: As pointed out above, to show that a countable union of countable sets is itself countable requires the Axiom of countable choice.
• There is a field with no algebraic closure.
• In all models of ZF¬C there is a vector space with no basis.
• There is a vector space with two bases of different cardinalities.
• There is a free complete Boolean algebra on countably many generators.^[40]
• There is a set that cannot be linearly ordered.
• There exists a model of ZF¬C in which every set in Rⁿ is
  measurable. Thus it is possible to exclude counterintuitive results like the Banach–Tarski paradox which are provable in ZFC. Furthermore, this is possible whilst assuming the Axiom of dependent choice, which is weaker than AC but sufficient to develop most of real analysis
  .
• In all models of ZF¬C, the
  generalized continuum hypothesis
  does not hold.

For proofs, see Jech (2008).

Additionally, by imposing definability conditions on sets (in the sense of descriptive set theory) one can often prove restricted versions of the axiom of choice from axioms incompatible with general choice. This appears, for example, in the Moschovakis coding lemma.

In type theory, a different kind of statement is known as the axiom of choice. This form begins with two types, σ and τ, and a relation R between objects of type σ and objects of type τ. The axiom of choice states that if for each x of type σ there exists a y of type τ such that R(x,y), then there is a function f from objects of type σ to objects of type τ such that R(x,f(x)) holds for all x of type σ:

${\displaystyle (\forall x^{\sigma })(\exists y^{\tau })R(x,y)\to (\exists f^{\sigma \to \tau })(\forall x^{\sigma })R(x,f(x)).}$

Unlike in set theory, the axiom of choice in type theory is typically stated as an

References

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• doi:10.4064/fm-46-1-1-13. Archived
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• Per Martin-Löf, "100 years of Zermelo's axiom of choice: What was the problem with it?", in Logicism, Intuitionism, and Formalism: What Has Become of Them?, Sten Lindström, Erik Palmgren, Krister Segerberg, and Viggo Stoltenberg-Hansen, editors (2008).
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• Herman Rubin, Jean E. Rubin: Equivalents of the Axiom of Choice II. North Holland/Elsevier, July 1985,
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• Ernst Zermelo, "Untersuchungen über die Grundlagen der Mengenlehre I," Mathematische Annalen 65: (1908) pp. 261–81. PDF download via digizeitschriften.de

Translated in:

ISBN 0-674-32449-8

• 1904. "Proof that every set can be well-ordered," 139-41.
• 1908. "Investigations in the foundations of set theory I," 199–215.

External links

Wikiquote has quotations related to Axiom of choice.

• Axiom of Choice entry in the Springer Encyclopedia of Mathematics.
• Axiom of Choice and Its Equivalents entry at ProvenMath. Includes formal statement of the Axiom of Choice, Hausdorff's Maximal Principle, Zorn's Lemma and formal proofs of their equivalence down to the finest detail.
• Consequences of the Axiom of Choice Archived 15 May 2021 at the Wayback Machine, based on the book by Paul Howard Archived 26 February 2021 at the Wayback Machine and Jean Rubin.
• "The Axiom of Choice" entry by John Lane Bell in the Stanford Encyclopedia of Philosophy.