Axiom of choice

This article is about the mathematical concept. For the band named after it, see Axiom of Choice (band).
(Si) is a family of sets indexed over the real numbers R; that is, there is a set Si for each real number i, with a small sample shown above. Each set contains a nonzero, and possibly infinite, number of elements. The axiom of choice allows us to arbitrarily select a single element from each set, forming a corresponding family of elements (xi) also indexed over the real numbers, with xi drawn from Si. In general the collections may be indexed over any set I, not just R.

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty. It states that for every indexed family of nonempty sets there exists an indexed family of elements such that for every . The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.[1]

Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin. In many cases such a selection can be made without invoking the axiom of choice; this is in particular the case if the number of bins is finite, or if a selection rule is available: a distinguishing property that happens to hold for exactly one object in each bin. To give an informal example, for any (even infinite) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection, but for an infinite collection of pairs of socks (assumed to have no distinguishing features), such a selection can be obtained only by invoking the axiom of choice.

Although originally controversial, the axiom of choice is now used without reservation by most mathematicians,[2] and it is included in the standard form of axiomatic set theory, Zermelo–Fraenkel set theory with the axiom of choice (ZFC). One motivation for this use is that a number of generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs. Contemporary set theorists also study axioms that are not compatible with the axiom of choice, such as the axiom of determinacy. The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced.

Statement

A choice function 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:

Axiom — For any set X of nonempty sets, there exists a choice function f defined on X.

Formally, this may be expressed as follows:

Thus, the negation of the axiom of choice states that there exists a collection of nonempty sets that has no choice function.

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.

Nomenclature ZF, AC, and ZFC

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

Variants

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 of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.[3]

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 be advised that this is a slightly different notion of choice function. Its domain is the powerset 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.[4]

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.

Restriction to finite sets

The 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 Zermelo–Fraenkel set theory without the axiom of choice (ZF); it is easily proved by mathematical induction.[5] 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.

Usage

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." In general, it is impossible to prove that F exists without the axiom of choice, but this seems to have gone unnoticed until Zermelo.

Not every situation requires the axiom of choice. For finite sets X, the axiom of choice follows from the other axioms of set theory. In that case it is equivalent to saying that if we have several (a finite number of) boxes, each containing at least one item, then we can choose exactly one item from each box. Clearly we can do this: We start at the first box, choose an item; go to the second box, choose an item; and so on. The number of boxes is finite, so eventually our choice procedure comes to an end. The result is an explicit choice function: a function that takes the first box to the first element we chose, the second box to the second element we chose, and so on. (A formal proof for all finite sets would use the principle of mathematical induction to prove "for every natural number k, every family of k nonempty sets has a choice function.") This method cannot, however, be used to show that every countable family of nonempty sets has a choice function, as is asserted by the axiom of countable choice. If the method is applied to an infinite sequence (Xi : i∈ω) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no "limiting" choice function can be constructed, in general, in ZF without the axiom of choice.

Examples

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 apply the axiom of choice.

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 set exists? 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. Namely, these are 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 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 select a point in each orbit requires the axiom of choice. See non-measurable set for more details.

The reason that we are able to choose least elements from subsets of the natural numbers is the fact that the natural numbers are well-ordered: every nonempty subset of the natural numbers has a unique least element under the natural ordering. 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.

Criticism and acceptance

A proof requiring the axiom of choice may establish the existence of an object without explicitly defining the object in the language of set theory. For example, while the axiom of choice implies that there is a well-ordering of the real numbers, there are models of set theory with the axiom of choice in which no well-ordering of the reals is definable. Similarly, although a subset of the real numbers that is not Lebesgue measurable can be proved to exist using the axiom of choice, it is consistent that no such set is definable.[6]

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.[7] 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.[8] 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.

Despite these seemingly paradoxical facts, most mathematicians accept the axiom of choice as a valid principle for proving new results in mathematics. The debate is interesting enough, however, that it is considered of note 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 which requires the axiom of choice to be true.

It is possible to prove many theorems using neither the axiom of choice nor its negation; such statements will be true in any model of Zermelo–Fraenkel set theory (ZF), regardless of the truth or falsity of the axiom of choice in that particular model. The restriction to ZF renders any claim that relies on either the axiom of choice or its negation unprovable. For example, the Banach–Tarski paradox 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. Similarly, all the statements listed below which require choice or some weaker version thereof for their proof are unprovable in ZF, but since each is provable in ZF plus the axiom of choice, there are models of ZF in which each statement is true. Statements such as the Banach–Tarski paradox 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.

In constructive mathematics

As discussed above, in ZFC, the axiom of choice is able to provide "nonconstructive proofs" in which the existence of an object is proved although no explicit example is constructed. ZFC, however, is still formalized in classical logic. The axiom of choice has also been thoroughly studied in the context of constructive mathematics, where non-classical logic is employed. The status of the axiom of choice varies between different varieties of constructive mathematics.

In Martin-Löf type theory and higher-order Heyting arithmetic, the appropriate statement of the axiom of choice is (depending on approach) included as an axiom or provable as a theorem.[9] Errett Bishop argued that the 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.[10]

In constructive set theory, however, Diaconescu's theorem shows that the axiom of choice implies the Law of excluded middle (unlike in Martin-Löf type theory, where it does not). Thus the axiom of choice is not generally available in constructive set theory. A cause for this difference is that the axiom of choice in type theory does not have the extensionality properties that the axiom of choice in constructive set theory does.[11]

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 in constructive set theory. Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.[12]

Independence

Assuming ZF is consistent, Kurt Gödel showed that the negation of the axiom of choice is not a theorem of ZF by constructing an inner model (the constructible universe) which satisfies ZFC and thus showing that ZFC is consistent. Assuming ZF is consistent, Paul Cohen employed the technique of forcing, developed for this purpose, to show that the axiom of choice itself is not a theorem of ZF by constructing a much more complex model which satisfies ZF¬C (ZF with the negation of AC added as axiom) and thus showing that ZF¬C is consistent. 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. The decision must be made on other grounds.

One argument given in favor of using the axiom of choice is that it is convenient to use it because it allows one to prove some simplifying propositions that otherwise could not be proved. Many theorems which are provable using choice are of an elegant general character: every ideal in a ring is contained in a maximal ideal, every vector space has a basis, and every product of compact spaces is compact. Without the axiom of choice, these theorems may not hold for mathematical objects of large cardinality.

The proof of the independence result also shows that a wide class of mathematical statements, including all statements that can be phrased in the language of Peano arithmetic, are provable in ZF if and only if they are provable in ZFC.[13] Statements in this class include the statement that P = NP, the Riemann hypothesis, and many other unsolved mathematical problems. When one attempts to solve problems in this class, it makes no difference whether ZF or ZFC is employed if the only question is the existence of a proof. It is possible, however, that there is a shorter proof of a theorem from ZFC than from ZF.

The axiom of choice is not the only significant statement which 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.

Stronger axioms

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 a possible axiom called the axiom of global choice which is stronger than the axiom of choice for sets because it also applies to proper classes. And the axiom of global choice follows from the axiom of limitation of size.

Equivalents

There are important statements that, assuming the axioms of ZF but neither AC nor ¬AC, are equivalent to the axiom of choice.[14] 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.

Category theory

There are several results in category theory which invoke the axiom of choice for their proof. These results might be weaker than, equivalent to, or stronger than the axiom of choice, depending on the strength of the technical foundations. For example, if one defines categories in terms of sets, that is, as sets of objects and morphisms (usually called a small category), or even locally small categories, whose hom-objects are sets, then there is no category of all sets, and so it is difficult for a category-theoretic formulation to apply to all sets. On the other hand, other foundational descriptions of category theory are considerably stronger, and an identical category-theoretic statement of choice may be stronger than the standard formulation, à la class theory, mentioned above.

Examples of category-theoretic statements which require choice include:

Weaker forms

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 (ACω 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.

Other choice axioms weaker than axiom of choice include the Boolean prime ideal theorem and the axiom of uniformization. The former is equivalent in ZF to the existence of an ultrafilter containing each given filter, proved by Tarski in 1930.

Results requiring AC (or weaker forms) but weaker than it

One of the most interesting aspects of the axiom of choice is the large number of places in mathematics that 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.

Stronger forms of the negation of AC

Now, consider stronger forms of the negation of AC. For example, 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. Note that strengthened negations may be compatible with weakened forms of AC. For example, ZF + DC[22] + BP is consistent, if ZF is.

It is also consistent with ZF + DC that every set of reals is Lebesgue measurable; however, this consistency result, due to Robert M. Solovay, cannot be proved in ZFC itself, but requires a mild large cardinal assumption (the existence of an inaccessible cardinal). The much stronger axiom of determinacy, or AD, implies that every set of reals is Lebesgue measurable, has the property of Baire, and has the perfect set property (all three of these results are refuted by AC itself). ZF + DC + AD is consistent provided that a sufficiently strong large cardinal axiom is consistent (the existence of infinitely many Woodin cardinals).

Statements consistent with the negation of AC

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 prove the negation of some standard facts. Note that any model of ZF¬C is also a model of ZF, so for each of the following statements, there exists a model of ZF in which that statement is true.

For proofs, see Jech (2008).

Quotes

"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" — Jerry Bona[25]

This is a joke: although the three are all mathematically equivalent, many mathematicians find the axiom of choice to be intuitive, the well-ordering principle to be counterintuitive, and Zorn's lemma to be too complex for any intuition.

"The Axiom of Choice is necessary to select a set from an infinite number of pairs of socks, but not an infinite number of pairs of shoes." — Bertrand Russell[26]

The observation here is that one can define a function to select from an infinite number of pairs of shoes by stating for example, to choose a left shoe. Without the axiom of choice, one cannot assert that such a function exists for pairs of socks, because left and right socks are (presumably) indistinguishable.

"Tarski tried to publish his theorem [the equivalence between AC and 'every infinite set A has the same cardinality as AxA', see above] in Comptes Rendus, but Fréchet and Lebesgue refused to present it. Fréchet wrote that an implication between two well known [true] propositions is not a new result, and Lebesgue wrote that an implication between two false propositions is of no interest".

Polish-American mathematician Jan Mycielski relates this anecdote in a 2006 article in the Notices of the AMS.[27]

"The axiom gets its name not because mathematicians prefer it to other axioms." — A. K. Dewdney

This quote comes from the famous April Fools' Day article in the computer recreations column of the Scientific American, April 1989.

Notes

  1. ↑ Zermelo 1904.
  2. ↑ Jech, 1977, p. 348ff; Martin-Löf 2008, p. 210. According to Mendelson 1964, p. 201:
    The status of the Axiom of Choice has become less controversial in recent years. To most mathematicians it seems quite plausible and it has so many important applications in practically all branches of mathematics that not to accept it would seem to be a wilful hobbling of the practicing mathematician.
  3. ↑ Herrlich 2006, p. 9. According to Suppes 1972, p. 243, this was the formulation of the axiom of choice which was originally given by Zermelo 1904. See also Halmos 1960, p. 60 for this formulation.
  4. ↑ Suppes 1972, p. 240.
  5. ↑ Tourlakis (2003), pp. 209–210, 215–216.
  6. ↑ Fraenkel, Abraham A.; Bar-Hillel, Yehoshua; Lévy, Azriel (1973), Foundations of set theory (2nd ed.), Amsterdam-London: North-Holland Publishing Co., pp. 69–70, MR 0345816.
  7. ↑ Rosenbloom, Paul C. (2005), The Elements of Mathematical Logic, Courier Dover Publications, p. 147, ISBN 9780486446172.
  8. ↑ Dawson, J.W. (August 2006), "Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödel's Impact on Logic, Mathematics, and Computer Science", Proc. 21st Annual IEEE Symposium on Logic in Computer Science (LICS 2006), pp. 339–341, doi:10.1109/LICS.2006.47, The axiom of choice, though it had been employed unconsciously in many arguments in analysis, became controversial once made explicit, not only because of its non-constructive character, but because it implied such extremely unintuitive consequences as the Banach- Tarski paradox..
  9. ↑ Per Martin-Löf, Intuitionistic type theory, 1980. Anne Sjerp Troelstra, Metamathematical investigation of intuitionistic arithmetic and analysis, Springer, 1973.
  10. ↑ Errett Bishop and Douglas S. Bridges, Constructive analysis, Springer-Verlag, 1985.
  11. ↑ Martin-Löf, Per (2006). "100 Years of Zermelo's Axiom of Choice: What was the Problem with It?". The Computer Journal. 49 (3): 345–350. doi:10.1093/comjnl/bxh162.
  12. ↑ Fred Richman, “Constructive mathematics without choice”, in: Reuniting the Antipodes—Constructive and Nonstandard Views of the Continuum (P. Schuster et al., eds), Synthèse Library 306, 199–205, Kluwer Academic Publishers, Amsterdam, 2001.
  13. ↑ This is because arithmetical statements are absolute to the constructible universe L. Shoenfield's absoluteness theorem gives a more general result.
  14. ↑ See Moore 2013, pp. 330–334, for a structured list of 74 equivalents. See Howard & Rubin 1998, pp. 11–16, for 86 equivalents with source references.
  15. ↑ Blass, Andreas (1984). "Existence of bases implies the axiom of choice". Contemporary mathematics. 31.
  16. ↑ A. Hajnal, A. Kertész: Some new algebraic equivalents of the axiom of choice, Publ. Math. Debrecen, 19(1972), 339–340, see also H. Rubin, J. Rubin, Equivalents of the axiom of choice, II, North-Holland, 1985, p. 111.
  17. ↑ Serre, Jean-Pierre (2003), Trees, Springer Monographs in Mathematics, Springer, p. 23; Soukup, Lajos (2008), "Infinite combinatorics: from finite to infinite", Horizons of combinatorics, Bolyai Soc. Math. Stud., 17, Berlin: Springer, pp. 189–213, doi:10.1007/978-3-540-77200-2_10, MR 2432534. See in particular Theorem 2.1, pp. 192–193.
  18. ↑ It is shown by Jech 2008, pp. 119–131, that the axiom of countable choice implies the equivalence of infinite and Dedekind-infinite sets, but that the equivalence of infinite and Dedekind-infinite sets does not imply the axiom of countable choice in ZF.
  19. ↑ It was shown by Lévy 1958 and others using Mostowski models that eight definitions of a finite set are independent in ZF without AC, although they are equivalent when AC is assumed. The definitions are I-finite, Ia-finite, II-finite, III-finite, IV-finite, V-finite, VI-finite and VII-finite. I-finiteness is the same as normal finiteness. IV-finiteness is the same as Dedekind-finiteness.
  20. ↑ "[FOM] Are (C,+) and (R,+) isomorphic".
  21. ↑ http://journals.cambridge.org/action/displayFulltext?type=1&fid=4931240&aid=4931232
  22. ↑ Axiom of dependent choice
  23. ↑ Jech 2008, pp. 142–144, Theorem 10.6 with proof.
  24. ↑ Stavi, Jonathan (1974). "A model of ZF with an infinite free complete Boolean algebra" (reprint). Israel Journal of Mathematics. 20 (2): 149–163. doi:10.1007/BF02757883.
  25. ↑ Krantz, Steven G. (2002), "The axiom of choice", Handbook of Logic and Proof Techniques for Computer Science, Springer, pp. 121–126, doi:10.1007/978-1-4612-0115-1_9.
  26. ↑ The boots-and-socks metaphor was given in 1919 by Russell 1993, pp. 125–127. He suggested that a millionaire might have ℵ0 pairs of boots and ℵ0 pairs of socks.
    "Among boots we can distinguish right and left, and therefore we can make a selection of one out of each pair, namely, we can choose all the right boots or all the left boots; but with socks no such principle of selection suggests itself, and we cannot be sure, unless we assume the multiplicative axiom, that there is any class consisting of one sock out of each pair."
    Russell generally used the term "multiplicative axiom" for the axiom of choice. Referring to the ordering of a countably infinite set of pairs of objects, he wrote:
    "There is no difficulty in doing this with the boots. The pairs are given as forming an ℵ0, and therefore as the field of a progression. Within each pair, take the left boot first and the right second, keeping the order of the pair unchanged; in this way we obtain a progression of all the boots. But with the socks we shall have to choose arbitrarily, with each pair, which to put first; and an infinite number of arbitrary choices is an impossibility. Unless we can find a rule for selecting, i.e. a relation which is a selector, we do not know that a selection is even theoretically possible."
    Russell then suggests using the location of the centre of mass of each sock as a selector.
  27. ↑ Mycielski, Jan (2006), "A system of axioms of set theory for the rationalists" (PDF), Notices of the American Mathematical Society, 53 (2): 206–213, MR 2208445.

References

Translated in: Jean van Heijenoort, 2002. From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. New edition. Harvard University Press. 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

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