Axiom of choice - Knowledge

1193:: 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. 7254: 4913: 65: 1264:

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.

29: 350:. 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. 346:

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

2019:, 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. 1237:, 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 4098:

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

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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

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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

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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

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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

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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

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is compact, among many others. Frequently, the axiom of choice allows generalizing a theorem to "larger" objects. For example, it is provable without the axiom of choice that every vector space of finite dimension has a basis, but the generalization to all vector spaces requires the axiom of choice.

1513:, developed for this purpose, to show that, assuming ZF is consistent, the axiom of choice itself is not a theorem of ZF. He did this by constructing a much more complex model that satisfies ZF¬C (ZF with the negation of AC added as axiom) and thus showing that ZF¬C is consistent. Cohen's model is a 1078:

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

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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

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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

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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

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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

1581:, and many other unsolved mathematical problems. When attempting 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. 3359:

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

1720:: 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.) 3549:

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

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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

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is represented as a marble on the right. Colors are used to suggest a functional association of marbles after adopting the choice axiom. The existence of such a choice function is in general independent of ZF for collections of infinite cardinality, even if all

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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

2892:, 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 + BP is consistent, if ZF is. 1082:

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

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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

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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

1256:(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. 4085:, 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. 3661:

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,

95:, with a small sample shown above. Each set contains at least one, and possibly infinitely many, elements. The axiom of choice allows us to select a single element from each set, forming a corresponding family of elements ( 3011:

The real numbers are a countable union of countable sets. 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

165:. 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 1221:

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. Because there is no

1303:, the appropriate statement of the axiom of choice is (depending on approach) included as an axiom or provable as a theorem. A cause for this difference is that the axiom of choice in type theory does not have the 774:

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.

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collection of pairs of socks (assumed to have no distinguishing features), 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.

2950:: 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 2820:. 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. 1073:

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

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of the real numbers, there are models of set theory with the axiom of choice in which no individual well-ordering of the reals is definable. Similarly, although a subset of the real numbers that is not

3150: 1588:(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. 2919:(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 2627:

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.

2221: 2933:(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. 2867: 588:

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

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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.

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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

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Dawson, J. W. (August 2006), "Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödel's Impact on Logic, Mathematics, and Computer Science",

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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

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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

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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

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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

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principle implies AC is the oldest open problem in set theory, and the equivalences of the other statements are similarly hard old open problems. In every

377:, 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 5633: 2073:, 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. 1103:. 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 1008:

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

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Although originally controversial, the axiom of choice is now used without reservation by most mathematicians, and is included in the standard form of

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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

1182:, 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 1562:

Likewise, a finite product of compact spaces can be proven to be compact without the axiom of choice, but the generalization to infinite products (

1174:. The set of those translates partitions the circle into a countable collection of pairwise disjoint sets, which are all pairwise congruent. Since 3080:) one can often prove restricted versions of the axiom of choice from axioms incompatible with general choice. This appears, for example, in the 2942:

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.

6308: 5370: 1605: 4580: 4820: 1750:: Every partially ordered set has a maximal chain. Equivalently, in any partially ordered set, every chain can be extended to a maximal chain. 1099:

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

1323:, who is notable for developing a framework for constructive analysis, argued that an axiom of choice was constructively acceptable, saying 1233:

Another argument against the axiom of choice is that it implies the existence of objects that may seem counterintuitive. One example is the

6391: 5532: 3767: 1517:, which is similar to permutation models, but uses "generic" subsets of the natural numbers (justified by forcing) in place of urelements. 1616:

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

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

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in which the existence of a type of object is proved without an explicit instance being constructed. In fact, in set theory and

3290:{\displaystyle (\forall x^{\sigma })(\exists y^{\tau })R(x,y)\to (\exists f^{\sigma \to \tau })(\forall x^{\sigma })R(x,f(x)).} 2057:

There are several weaker statements that are not equivalent to the axiom of choice but are closely related. One example is the

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properties that the axiom of choice in constructive set theory does. The type theoretical context is discussed further below.

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with the axiom of choice (ZFC). One motivation for this is that a number of generally accepted mathematical results, such as

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is well-orderable. As the ordinal parameter is increased, these approximate the full axiom of choice more and more closely.

6718: 6041: 5059: 4879: 3803: 4811: 3096:, a different kind of statement is known as the axiom of choice. This form begins with two types, σ and τ, and a relation 1651:. In fact, Zermelo initially introduced the axiom of choice in order to formalize his proof of the well-ordering theorem. 4850: 6723: 6713: 6450: 6303: 5656: 5387: 3959: 3880: 1621: 5647: 2624: 2611: 1717: 6859: 4779: 4704: 4607: 4596: 4539: 4492: 4394: 3792: 3501: 3466: 3383: 1689:: If two sets are given, then either they have the same cardinality, or one has a smaller cardinality than the other. 1521: 6201: 2782: 6956: 6700: 5525: 4515: 4355: 3885: 3062: 2730:

model of ZF where choice fails, these statements fail too, but it is unknown whether they can hold without choice.

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Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a

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

1573:, are provable in ZF if and only if they are provable in ZFC. Statements in this class include the statement that 6261: 5954: 5365: 2178: 1710:

of any family of nonempty sets is nonempty. In other words, every family of nonempty sets has a choice function (

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

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sets in the family. The axiom of choice asserts the existence of such elements; it is therefore equivalent to:

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There is no infinite decreasing sequence of cardinals. The equivalence was conjectured by Schoenflies in 1905.

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the object in the language of set theory. For example, while the axiom of choice implies that there is a

591: 2915:, or AD, implies that every set of reals is Lebesgue measurable, has the property of Baire, and has the 2152: 1792:, a linearly independent spanning subset). In other words, vector spaces are equivalent to free modules. 1138:

consisting of all rational rotations, that is, rotations by angles which are rational multiples of

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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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The axiom of choice is not the only significant statement that is independent of ZF. For example, the

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

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It has been known since as early as 1922 that the axiom of choice may fail in a variant of ZF with

919:, whereas with the definition used elsewhere in this article, the domain of a choice function on a 4817: 4487:, Sten Lindström, Erik Palmgren, Krister Segerberg, and Viggo Stoltenberg-Hansen, editors (2008). 1201:

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

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but neither AC nor ¬AC, are equivalent to the axiom of choice. The most important among them are

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each imply the axiom of choice and so are strictly stronger than it. In class theories such as

1563: 1542: 1505:) that satisfies ZFC, thus showing that ZFC is consistent if ZF itself is consistent. In 1963, 1502: 1285: 1245: 1183: 1104: 1013: 374: 20: 4054: 3951: 3491: 3373: 1806:. Equivalently, in any nontrivial unital ring, every ideal can be extended to a maximal ideal. 911:, but this is a slightly different notion of choice function. Its domain is the power set of 7127: 6980: 6772: 6490: 6226: 6132: 5991: 5976: 5857: 5832: 5352: 5322: 5266: 5186: 5166: 5144: 3456: 3405:, p. 243, this was the formulation of the axiom of choice which was originally given by 3100:

between objects of type σ and objects of type τ. The axiom of choice states that if for each

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Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III

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Per Martin-Löf, "100 years of Zermelo's axiom of choice: What was the problem with it?", in

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Stavi, Jonathan (1974). "A model of ZF with an infinite free complete Boolean algebra".

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Sageev, Gershon (March 1975). "An independence result concerning the axiom of choice".

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

2828: 2598: 2045: 2016: 1900: 1867: 1852: 1814: 1775: 1757: 1753: 1744:, totally ordered subset) has an upper bound contains at least one maximal element. 1570: 1554: 1354: 1211: 353: 3982: 3863: 2015:), or even locally small categories, whose hom-objects are sets, then there is no 1178:

is not measurable for any rotation-invariant countably additive finite measure on

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

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exists without the axiom of choice, but this seems to have gone unnoticed until

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Proceedings of the National Academy of Sciences of the United States of America

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is well-orderable. Given an ordinal parameter α ≥ 1 — for every set

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which are provable in ZFC. Furthermore, this is possible whilst assuming the

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There is an infinite set of real numbers without a countably infinite subset.

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A uniform space is compact if and only if it is complete and totally bounded.

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

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varies over all formulas or over all formulas of a particular logical form.

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Additionally, by imposing definability conditions on sets (in the sense of

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ZFC – Zermelo–Fraenkel set theory, extended to include the Axiom of Choice.

166: 41:

represented as a jar and its elements represented as marbles. Each element

4309:(1922), "Der Begriff "definit" und die Unabhängigkeit des Auswahlaxioms", 2149:(with ZF) can be used to prove the Axiom of choice for finite sets: Given 7157: 7037: 6216: 6206: 6153: 5837: 5757: 5742: 5622: 5567: 5474: 5109: 4468: 4430: 4404: 4325: 4012:

Soukup, Lajos (2008), "Infinite combinatorics: from finite to infinite",

3833:

Blass, Andreas (1984). "Existence of bases implies the axiom of choice".

3526: 3519:

Proc. 21st Annual IEEE Symposium on Logic in Computer Science (LICS 2006)

3093: 2561: 2434:

There exist Lebesgue-measurable subsets of the real numbers that are not

1498: 1095:

were finite. If we try to choose an element from each set, then, because

1088: 134: 79: 4828: 3592:

Metamathematical investigation of intuitionistic arithmetic and analysis

2888:

If we abbreviate by BP the claim that every set of real numbers has the

2022:

Examples of category-theoretic statements which require choice include:

1714:

a function which maps each of the nonempty sets to one of its elements).

6087: 5942: 5913: 5719: 5454: 5225: 4888: 4730: 4587:: Equivalents of the axiom of choice. North Holland, 1963. Reissued by 4311:

Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften

4280: 3928: 3562: 3043: 2734: 2667: 2543: 2416: 2140: 1655: 1215: 1190: 1009: 154: 1386:, and build a model where each set is symmetric under the interchange 7239: 7142: 6195: 6112: 6072: 6036: 5972: 5784: 5774: 5747: 5510: 5257: 5220: 5171: 5069: 3866:, A. Kertész: Some new algebraic equivalents of the axiom of choice, 2435: 2376: 1767: 1763: 1346: 896: 729:. This is not the most general situation of a Cartesian product of a 3919: 3902: 3046:. Thus it is possible to exclude counterintuitive results like the 2785:: if two sets have surjections to each other, they are equinumerous. 7224: 7022: 6470: 6175: 5769: 4787:

1908. "Investigations in the foundations of set theory I," 199–215.

4768:

From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931

4588: 4564:

Zermelo's axiom of choice: Its origins, development & influence

2076:

Given an ordinal parameter α ≥ ω+2 — for every set

1033:

contains only non-empty sets, a mathematician might have said "let

585: 219: 3025:

There is a vector space with two bases of different cardinalities.

2589:

On every infinite-dimensional topological vector space there is a

1244:

Moreover, paradoxical consequences of the axiom of choice for the

823:∃c ∀e (e ∈ x → ∃a (a ∈ e ∧ a ∈ c ∧ ∀b ((b ∈ e ∧ b ∈ c) → a = b)))) 64: 28: 19:

This article is about the mathematical concept. For the band, see

6820: 5612: 4512:

Zermelo's axiom of choice, Its origins, development and influence

3819:, pp. 330–334, for a structured list of 74 equivalents. See 2457: 1957:

is included in a set that is maximal among consistent subsets of

1574: 1497:

of the axiom of choice is not a theorem of ZF by constructing an

1062: 2488:

contains an infinite linearly independent subset (this requires

1927:

of a product of subsets is equal to the product of the closures.

880:

containing exactly one element from each part of the partition.

5282: 5104: 4485:

Logicism, Intuitionism, and Formalism: What Has Become of Them?

1295:

The situation is different when the principle is formulated in

2895:

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

2136:

are true in all models of ZFC but false in some models of ZF.

1740:: Every non-empty partially ordered set in which every chain ( 6364: 5710: 5555: 5154: 4921: 4857: 4099:

finiteness. IV-finiteness is the same as Dedekind-finiteness.

2037:

If two small categories are weakly equivalent, then they are

1520:

Together these results establish that the axiom of choice is

915:(with the empty set removed), and so makes sense for any set 827:

Note that P ∨ Q ∨ R is logically equivalent to (¬P ∧ ¬Q) → R.

150: 4433:(1977). "About the Axiom of Choice". In John Barwise (ed.). 4184: 3823:, pp. 11–16, for 86 equivalents with source references. 3054:, which is weaker than AC but sufficient to develop most of 3022:

In all models of ZF¬C there is a vector space with no basis.

2103:

Other choice axioms weaker than axiom of choice include the

1692:

Given two non-empty sets, one has a surjection to the other.

1639:

There are important statements that, assuming the axioms of

3986: 2873:, where Ω is the set of Archimedean equivalence classes of 1995:. Equivalently, every nonempty graph has a spanning forest. 4748:, "Untersuchungen über die Grundlagen der Mengenlehre I," 4008:, Springer Monographs in Mathematics, Springer, p. 23 3950:(2nd ed.). Oxford: Oxford University Press. pp. 119:. In general, the collections may be indexed over any set 4784:

1904. "Proof that every set can be well-ordered," 139-41.

2936: 2720: 2131:

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

1624:

and states (in the vernacular) that every set belongs to

1361:. The basic technique can be illustrated as follows: Let 3042:

There exists a model of ZF¬C in which every set in R is

1214:

can be proved to exist using the axiom of choice, it is

4447:"The independence of various definitions of finiteness" 3896: 3894: 2883: 857:

such that its intersection with any of the elements of

802:

such that its intersection with any of the elements of

3443: 3124:

from objects of type σ to objects of type τ such that

2903:, cannot be proved in ZFC itself, but requires a mild 381:. The axiom of choice is avoided in some varieties of 4693:

Lectures in Logic and Set Theory. Vol. II: Set Theory

3677: 3153: 2846: 2812:

are equinumerous. Equivalently, a partition of a set

2700: 2676: 2640: 2361: 2334: 2233: 2181: 2155: 1766:

principle: Every partially ordered set has a maximal

1726:: Every set can be well-ordered. Consequently, every 1319:, which do not imply the law of the excluded middle. 820:∃a ∃b ∃c (a ∈ x ∧ b ∈ x ∧ c ∈ a ∧ c ∈ b ∧ ¬(a = b)) ∨ 657: 594: 464: 314: 274: 228: 178: 3891: 3665: 2961:

from the real numbers to the real numbers such that

1126:

Additionally, consider for instance the unit circle

967:

The negation of the axiom can thus be expressed as:

907:

Authors who use this formulation often speak of the

883:

Another equivalent axiom only considers collections

3903:"Injectivity, Projectivity and the Axiom of Choice" 2877:. This equivalence was conjectured by Hahn in 1907. 2496:

Stone's representation theorem for Boolean algebras

903:(with the empty set removed) has a choice function. 755:

AC – the Axiom of Choice. More rarely, AoC is used.

32:Illustration of the axiom of choice, with each set 4712:"Beweis, dass jede Menge wohlgeordnet werden kann" 4670: 4549:(1938), "Über den Begriff einer Endlichen Menge", 4378: 3565:The Axiom of Choice and the No-Signaling Principle 3289: 2861: 2706: 2682: 2646: 2367: 2347: 2262: 2215: 2167: 1154:breaks up into uncountably many orbits under 706: 643: 573: 397:(also called selector or selection) is a function 326: 300: 260: 210: 3907:Transactions of the American Mathematical Society 3563:Baumeler, Ä., Dakić, B. and Del Santo, F., 2022. 3087: 2840:order-embeds as a subgroup of the additive group 2423:, which states that there exists a subset of the 1189:In classical arithmetic, the natural numbers are 439:of nonempty sets, there exists a choice function 7270: 1760:has a maximal element with respect to inclusion. 1166:with the property that all of its translates by 810:This can be formalized in first-order logic as: 334:. The axiom of choice was formulated in 1904 by 4156:Journal of the Australian Mathematical Society 4137:"[FOM] Are (C,+) and (R,+) isomorphic" 2509:, that every subgroup of a free group is free. 1267: 1057:. In general, it is impossible to prove that 1016:. In the even simpler case of a collection of 887:that are essentially powersets of other sets: 425:. With this concept, the axiom can be stated: 163:of a collection of non-empty sets is non-empty 5526: 4873: 3352:; Martin-Löf 2008, p. 210. According to 1003: 829:In English, this first-order sentence reads: 4255:, pp. 142–144, Theorem 10.6 with proof. 3791:This is because arithmetical statements are 455:Formally, this may be expressed as follows: 4372: 4258: 3835:Axiomatic set theory (Boulder, Colo., 1983) 3820: 3496:, Courier Dover Publications, p. 147, 3019:There is a field with no algebraic closure. 2216:{\displaystyle \left(X_{i}\right)_{i\in I}} 1284:shows that the axiom of choice implies the 1248:in physics have recently been pointed out. 1196: 1087:is the set of all non-empty subsets of the 104:) also indexed over the real numbers, with 5718: 5533: 5519: 4880: 4866: 3603: 3489: 1525:theory. It must be made on other grounds. 489: 485: 405:of nonempty sets, such that for every set 4635: 4566:. Mineola, New York: Dover Publications. 4545: 4497: 4467: 4190: 4167: 4060: 4021: 3918: 3757:"The Independence of the Axiom of Choice" 3731: 3721: 3683: 3371: 3353: 2849: 1916:: The Cartesian product of any family of 1802:(other than the trivial ring) contains a 1292:, where non-classical logic is employed. 1288:. The principle is thus not available in 567: 537: 4641:Handbook of Analysis and Its Foundations 4346: 4305: 3671: 3398: 2788:Weak partition principle: if there is a 1631:, is stronger than the axiom of choice. 1107:(0,1) does not have a least element: if 786:, if the empty set is not an element of 63: 27: 4709: 4619:Introduction to mathematical philosophy 4613: 4354:. Lecture Notes in Math. 1876. Berlin: 4221: 3434:Tourlakis (2003), pp. 209–210, 215–216. 3406: 3324: 1977:; see the section "Weaker forms" below. 1831:group structure and the axiom of choice 843:contains the empty set as an element or 338:in order to formalize his proof of the 7271: 5540: 4665: 4324: 4152:"A consequence of the axiom of choice" 4108: 4069:Topology for the Working Mathematician 4011: 3941: 3881:Equivalents of the axiom of choice, II 3516: 3422: 3410: 3402: 3372:Rosenberg, Steven (21 December 2021). 2937:Statements implying the negation of AC 2899:, but this consistency result, due to 2862:{\displaystyle \mathbb {R} ^{\Omega }} 2721:Possibly equivalent implications of AC 1881:The closed unit ball of the dual of a 1091:. First we might try to proceed as if 721:of nonempty sets is an element of the 5514: 4861: 4754:PDF download via digizeitschriften.de 4561: 4509: 4264: 4066: 4000: 3900: 3832: 3816: 3754: 3695: 3037:a set that cannot be linearly ordered 2111:. The former is equivalent in ZF to 1457:can be in the model but sets such as 1341:List of statements independent of ZFC 717:Each choice function on a collection 169:. Formally, it states that for every 4479:from the original on 9 October 2022. 4441: 4429: 4403: 4252: 4095: 4082: 3874:(1972), 339–340, see also H. Rubin, 3567:, arXiv preprint — arXiv:2206.08467. 3336: 3070: 2884:Stronger forms of the negation of AC 2578:, and its consequences, such as the 2492:, but not the full axiom of choice). 2263:{\displaystyle \prod _{i\in I}X_{i}} 2061:(DC). A still weaker example is the 2048:(the Freyd adjoint functor theorem). 1606:Von Neumann–Bernays–Gödel set theory 1079:choice to our axioms of set theory. 979:(on the set of non-empty subsets of 4851:Stanford Encyclopedia of Philosophy 4818:Consequences of the Axiom of Choice 4812:Axiom of Choice and Its Equivalents 4381:Consequences of the axiom of choice 4233:Stanford Encyclopedia of Philosophy 4149: 3773:from the original on 9 October 2022 2929:'s system of axiomatic set theory, 2739:Partition principle: if there is a 1620:. Tarski's axiom, which is used in 1119:/2 is always strictly smaller than 943:such that for any non-empty subset 644:{\displaystyle \neg \forall X\left} 13: 4505:. New York: Van Nostrand Reinhold. 4503:Introduction to Mathematical Logic 3493:The Elements of Mathematical Logic 3413:, p. 60 for this formulation. 3241: 3216: 3176: 3157: 2854: 2701: 2677: 2641: 2285:but not the full axiom of choice). 2168:{\displaystyle I\neq \varnothing } 2001: 1591: 684: 658: 598: 595: 525: 490: 465: 222:sets, there exists an indexed set 14: 7290: 4795: 4643:. San Diego, CA: Academic Press. 4053:. See in particular Theorem 2.1, 3804:Shoenfield's absoluteness theorem 3375:An Invitation to Abstract Algebra 2162: 1969:first-order sentences in a given 574:{\displaystyle \forall X\left\,.} 157:equivalent to the statement that 7252: 4911: 4621:. New York: Dover Publications. 3063:generalized continuum hypothesis 2907:assumption (the existence of an 1829:binary operation is enough, see 1756:: Every non-empty collection of 1602:generalized continuum hypothesis 1586:generalized continuum hypothesis 1566:) requires the axiom of choice. 1259:Theorems of ZF hold true in any 927:Every set has a choice function. 261:{\displaystyle (x_{i})_{i\in I}} 211:{\displaystyle (S_{i})_{i\in I}} 4246: 4210: 4196: 4143: 4129: 4102: 4088: 4075: 3994: 3976: 3935: 3857: 3826: 3809: 3785: 3748: 3696:Gödel, Kurt (9 November 1938). 3689: 3655: 3635: 3597: 3571: 3556: 3510: 2816:cannot be strictly larger than 2052: 1612:, there is an axiom called the 1486:cannot have a choice function. 1404:for all but a finite number of 1334: 1218:that no such set is definable. 746: 524: 4887: 4435:Handbook of Mathematical Logic 3483: 3437: 3428: 3416: 3392: 3365: 3342: 3330: 3318: 3281: 3278: 3272: 3260: 3254: 3238: 3235: 3227: 3213: 3210: 3207: 3195: 3189: 3173: 3170: 3154: 3088:Axiom of choice in type theory 2836:: Every ordered abelian group 1920:topological spaces is compact. 1634: 1622:Tarski–Grothendieck set theory 1123:. So this attempt also fails. 707:{\displaystyle \exists X\left} 696: 690: 678: 672: 633: 627: 621: 618: 612: 559: 550: 544: 538: 502: 486: 301:{\displaystyle x_{i}\in S_{i}} 243: 229: 193: 179: 1: 7213:History of mathematical logic 4387:American Mathematical Society 4299: 4268:Israel Journal of Mathematics 3764:Stanford University Libraries 3032:on countably many generators. 1973:is weaker, equivalent to the 1923:In the product topology, the 1661:Tarski's theorem about choice 1014:principle of finite induction 861:contains exactly one element. 806:contains exactly one element. 762:omitting the Axiom of Choice. 7138:Primitive recursive function 4677:. Mineola, New York: Dover. 4204:"On the Partition Principle" 4123:10.1016/0003-4843(75)90002-9 4111:Annals of Mathematical Logic 4032:10.1007/978-3-540-77200-2_10 3806:gives a more general result. 3490:Rosenbloom, Paul C. (2005), 3120:), then there is a function 2625:Gödel's completeness theorem 2542:, allowing the extension of 2281:is countable (this requires 2080:with rank less than α, 975:such that for all functions 850:are not pairwise disjoint or 388: 7: 4806:Encyclopedia of Mathematics 3061:In all models of ZF¬C, the 2612:Stone–Čech compactification 2500:Boolean prime ideal theorem 2484:Every infinite-dimensional 2277:of any countable family of 2105:Boolean prime ideal theorem 1975:Boolean prime ideal theorem 1748:Hausdorff maximal principle 1618:axiom of limitation of size 1379:be distinct urelements for 1349:, through the technique of 1268:In constructive mathematics 1068: 1041:) be one of the members of 769: 760:Zermelo–Fraenkel set theory 451:to an element of that set. 371:Zermelo–Fraenkel set theory 85:; that is, there is a set S 10: 7297: 6202:Schröder–Bernstein theorem 5929:Monadic predicate calculus 5588:Foundations of mathematics 5371:von Neumann–Bernays–Gödel 4752:: (1908) pp. 261–81. 4697:Cambridge University Press 4562:Moore, Gregory H (2013) . 4510:Moore, Gregory H. (1982). 3583:Intuitionistic type theory 3401:, p. 9. According to 2981:, i.e., for any sequence { 2783:Schröder–Bernstein theorem 1961:. The special case where 1949:is a consistent subset of 1509:employed the technique of 1338: 1004:Restriction to finite sets 872:the existence of a subset 817:∃o (o ∈ x ∧ ¬∃n (n ∈ o)) ∨ 798:, then there exists a set 401:, defined on a collection 18: 7248: 7235:Philosophy of mathematics 7184:Automated theorem proving 7166: 7061: 6893: 6786: 6638: 6355: 6331: 6309:Von Neumann–Bernays–Gödel 6254: 6148: 6052: 5950: 5941: 5868: 5803: 5709: 5631: 5548: 5435: 5398: 5310: 5200: 5172:One-to-one correspondence 5088: 5029: 4920: 4909: 4895: 4229:"Quine's New Foundations" 4217:Axiom of dependent choice 4169:10.1017/S1446788700031505 4014:Horizons of combinatorics 3458:Foundations of set theory 3104:of type σ there exists a 3052:Axiom of dependent choice 3014:Axiom of countable choice 2774:is less than or equal to 2553:has an orthonormal basis. 2063:axiom of countable choice 2059:axiom of dependent choice 1941:is a set of sentences of 1663:: For every infinite set 1598:axiom of constructibility 1357:and developed further by 1317:axiom of dependent choice 1313:axiom of countable choice 1299:. There and higher-order 78:of sets indexed over the 4834:26 February 2021 at the 4772:Harvard University Press 4750:Mathematische Annalen 65 3652:, Springer-Verlag, 1985. 3311: 3082:Moschovakis coding lemma 3030:complete Boolean algebra 2948:weak partition principle 2591:discontinuous linear map 2507:Nielsen–Schreier theorem 1847:Baer's criterion: Every 1809:For every non-empty set 1197:Criticism and acceptance 1111:is in (0,1), then so is 1024: 865:This guarantees for any 383:constructive mathematics 6885:Self-verifying theories 6706:Tarski's axiomatization 5657:Tarski's undefinability 5652:incompleteness theorems 4827:, based on the book by 4710:Zermelo, Ernst (1904). 4455:Fundamenta Mathematicae 4067:Muger, Michael (2020). 3901:Blass, Andreas (1979). 3843:10.1090/conm/031/763890 3821:Howard & Rubin 1998 3348:Jech, 1977, p. 348 2975:sequentially continuous 2707:{\displaystyle \Sigma } 2683:{\displaystyle \Sigma } 2647:{\displaystyle \Sigma } 2564:of sets of functionals. 2549:The theorem that every 2512:The additive groups of 2318:Eight definitions of a 2296:, then there exists an 2109:axiom of uniformization 1849:divisible abelian group 1610:Morse–Kelley set theory 1290:constructive set theory 1170:are disjoint from 1053:" to define a function 931:which is equivalent to 7259:Mathematics portal 6870:Proof of impossibility 6518:propositional variable 5828:Propositional calculus 5130:Constructible universe 4957:Constructibility (V=L) 4804:entry in the Springer 3942:Awodey, Steve (2010). 3797:constructible universe 3723:10.1073/pnas.24.12.556 3291: 3078:descriptive set theory 2863: 2834:Hahn embedding theorem 2763:. Equivalently, every 2708: 2684: 2648: 2569:Baire category theorem 2558:Banach–Alaoglu theorem 2369: 2349: 2264: 2217: 2169: 1885:over the reals has an 1614:axiom of global choice 1503:constructible universe 1329: 1297:Martin-Löf type theory 1286:law of excluded middle 1274:nonconstructive proofs 1246:no-signaling principle 1150:is uncountable. Hence 708: 645: 575: 328: 327:{\displaystyle i\in I} 302: 262: 212: 130: 61: 21:Axiom of Choice (band) 7128:Kolmogorov complexity 7081:Computably enumerable 6981:Model complete theory 6773:Principia Mathematica 5833:Propositional formula 5662:Banach–Tarski paradox 5353:Principia Mathematica 5187:Transfinite induction 5046:(i.e. set difference) 4842:"The Axiom of Choice" 4719:Mathematische Annalen 4411:. Mineola, New York: 3650:Constructive analysis 3629:10.1093/comjnl/bxh162 3292: 3048:Banach–Tarski paradox 2965:is not continuous at 2911:). The much stronger 2909:inaccessible cardinal 2871:lexicographical order 2864: 2709: 2685: 2649: 2450:Banach–Tarski paradox 2370: 2350: 2348:{\displaystyle G_{S}} 2265: 2218: 2170: 2071:mathematical analysis 1724:Well-ordering theorem 1649:well-ordering theorem 1629:Grothendieck universe 1522:logically independent 1325: 1235:Banach–Tarski paradox 709: 646: 576: 447:and maps each set of 340:well-ordering theorem 329: 303: 263: 213: 91:for each real number 67: 31: 7076:Church–Turing thesis 7063:Computability theory 6272:continuum hypothesis 5790:Square of opposition 5648:Gödel's completeness 5427:Burali-Forti paradox 5182:Set-builder notation 5135:Continuum hypothesis 5075:Symmetric difference 4673:Axiomatic set theory 4469:10.4064/fm-46-1-1-13 3888:, 1985, p. 111. 3868:Publ. Math. Debrecen 3755:Cohen, Paul (2019). 3609:The Computer Journal 3588:Anne Sjerp Troelstra 3527:10.1109/LICS.2006.47 3521:, pp. 339–341, 3449:Bar-Hillel, Yehoshua 3445:Fraenkel, Abraham A. 3151: 3108:of type τ such that 2957:There is a function 2946:The negation of the 2917:perfect set property 2913:axiom of determinacy 2844: 2698: 2674: 2638: 2584:closed graph theorem 2580:open mapping theorem 2419:on the existence of 2359: 2332: 2231: 2227:sets, their product 2179: 2153: 2123:is a subset of some 2017:category of all sets 1282:Diaconescu's theorem 1254:logically equivalent 1130:, and the action on 939:there is a function 909:choice function on A 790:and the elements of 655: 592: 462: 379:axiom of determinacy 367:axiomatic set theory 312: 272: 226: 176: 7230:Mathematical object 7121:P versus NP problem 7086:Computable function 6880:Reverse mathematics 6806:Logical consequence 6683:primitive recursive 6678:elementary function 6451:Free/bound variable 6304:Tarski–Grothendieck 5823:Logical connectives 5753:Logical equivalence 5603:Logical consequence 5388:Tarski–Grothendieck 4823:15 May 2021 at the 4764:Jean van Heijenoort 4409:The axiom of choice 4193:, pp. 391–392. 4150:Ash, C. J. (1975). 4139:. 21 February 2006. 3714:1938PNAS...24..556G 3621:1980CompJ..23..262L 2897:Lebesgue measurable 2658:(or alternatively, 2632:compactness theorem 2540:functional analysis 2536:Hahn–Banach theorem 2530:Functional analysis 2479:transcendence basis 2429:Lebesgue measurable 2421:non-measurable sets 2006:Several results in 1914:Tychonoff's theorem 1883:normed vector space 1877:Functional analysis 1697:surjective function 1564:Tychonoff's theorem 1239:non-measurable sets 1212:Lebesgue measurable 1146:is countable while 853:there exists a set 443:that is defined on 433: — 421:) is an element of 375:Tychonoff's theorem 16:Axiom of set theory 7028:Transfer principle 6991:Semantics of logic 6976:Categorical theory 6952:Non-standard model 6466:Logical connective 5593:Information theory 5542:Mathematical logic 4977:Limitation of size 4731:10.1007/BF01445300 4691:George Tourlakis, 4547:Mostowski, Andrzej 4532:Dover Publications 4499:Mendelson, Elliott 4413:Dover Publications 4281:10.1007/BF02757883 4002:Serre, Jean-Pierre 3646:Douglas S. Bridges 3287: 2859: 2704: 2680: 2644: 2620:Mathematical logic 2544:linear functionals 2365: 2345: 2260: 2249: 2213: 2165: 1933:Mathematical logic 1908:topological spaces 1895:Point-set topology 1838:free abelian group 1671:between the sets 1579:Riemann hypothesis 1351:permutation models 1301:Heyting arithmetic 1186:for more details. 1184:non-measurable set 995:) does not lie in 921:collection of sets 867:partition of a set 704: 641: 571: 520: 431: 324: 298: 258: 208: 131: 62: 7266: 7265: 7198:Abstract category 7001:Theories of truth 6811:Rule of inference 6801:Natural deduction 6782: 6781: 6327: 6326: 6032:Cartesian product 5937: 5936: 5843:Many-valued logic 5818:Boolean functions 5701:Russell's paradox 5676:diagonal argument 5573:First-order logic 5508: 5507: 5417:Russell's paradox 5366:Zermelo–Fraenkel 5267:Dedekind-infinite 5140:Diagonal argument 5039:Cartesian product 4903:Set (mathematics) 4684:978-0-486-61630-8 4650:978-0-12-622760-4 4628:978-0-486-27724-0 4615:Russell, Bertrand 4573:978-0-486-48841-7 4530:, available as a 4525:978-0-387-90670-6 4422:978-0-486-46624-8 4365:978-3-540-30989-5 4307:Fraenkel, Abraham 4041:978-3-540-77199-9 3983:projective object 3594:, Springer, 1973. 3536:978-0-7695-2631-7 3140:)) holds for all 2901:Robert M. Solovay 2890:property of Baire 2468:algebraic closure 2443:Hausdorff paradox 2368:{\displaystyle S} 2313:Dedekind infinite 2234: 2175:and a collection 2147:ultrafilter lemma 2117:ultrafilter lemma 1943:first-order logic 1903:of any family of 1901:Cartesian product 1860:projective object 1708:Cartesian product 1533:with unity has a 1482:cannot, and thus 1359:Andrzej Mostowski 1010:finite collection 796:pairwise disjoint 723:Cartesian product 505: 429: 161:Cartesian product 74:) is an infinite 7286: 7257: 7256: 7208:History of logic 7203:Category of sets 7096:Decision problem 6875:Ordinal analysis 6816:Sequent calculus 6714:Boolean algebras 6654: 6653: 6628: 6599:logical/constant 6353: 6352: 6339: 6262:Zermelo–Fraenkel 6013:Set operations: 5948: 5947: 5885: 5716: 5715: 5696:Löwenheim–Skolem 5583:Formal semantics 5535: 5528: 5521: 5512: 5511: 5490:Bertrand Russell 5480:John von Neumann 5465:Abraham Fraenkel 5460:Richard Dedekind 5422:Suslin's problem 5333:Cantor's theorem 5050:De Morgan's laws 4915: 4882: 4875: 4868: 4859: 4858: 4742: 4716: 4688: 4676: 4662: 4632: 4577: 4558: 4529: 4506: 4480: 4478: 4471: 4451: 4438: 4426: 4400: 4384: 4369: 4343: 4331:Naive Set Theory 4321: 4293: 4292: 4262: 4256: 4250: 4244: 4243: 4241: 4239: 4225: 4219: 4214: 4208: 4207: 4200: 4194: 4188: 4182: 4181: 4171: 4147: 4141: 4140: 4133: 4127: 4126: 4106: 4100: 4094:It was shown by 4092: 4086: 4079: 4073: 4072: 4064: 4058: 4055:pp. 192–193 4052: 4025: 4009: 3998: 3992: 3980: 3974: 3973: 3949: 3939: 3933: 3932: 3922: 3898: 3889: 3861: 3855: 3854: 3830: 3824: 3813: 3807: 3789: 3783: 3782: 3780: 3778: 3772: 3761: 3752: 3746: 3745: 3735: 3725: 3693: 3687: 3681: 3675: 3669: 3663: 3659: 3653: 3639: 3633: 3632: 3601: 3595: 3575: 3569: 3560: 3554: 3552: 3514: 3508: 3506: 3487: 3481: 3479: 3441: 3435: 3432: 3426: 3420: 3414: 3396: 3390: 3389: 3369: 3363: 3346: 3340: 3334: 3328: 3322: 3296: 3294: 3293: 3288: 3253: 3252: 3234: 3233: 3188: 3187: 3169: 3168: 3069:For proofs, see 3028:There is a free 2988:} converging to 2921:Woodin cardinals 2868: 2866: 2865: 2860: 2858: 2857: 2852: 2829:Abstract algebra 2713: 2711: 2710: 2705: 2689: 2687: 2686: 2681: 2666:such that every 2653: 2651: 2650: 2645: 2599:General topology 2490:dependent choice 2374: 2372: 2371: 2366: 2354: 2352: 2351: 2346: 2344: 2343: 2283:countable choice 2269: 2267: 2266: 2261: 2259: 2258: 2248: 2222: 2220: 2219: 2214: 2212: 2211: 2200: 2196: 2195: 2174: 2172: 2171: 2166: 2092:less than ω 1821:that gives it a 1815:binary operation 1798:: Every unital 1776:Abstract algebra 1758:finite character 1571:Peano arithmetic 1493:showed that the 1481: 1456: 1385: 1355:Abraham Fraenkel 846:the elements of 713: 711: 710: 705: 703: 699: 650: 648: 647: 642: 640: 636: 580: 578: 577: 572: 566: 562: 519: 434: 354:Bertrand Russell 333: 331: 330: 325: 307: 305: 304: 299: 297: 296: 284: 283: 267: 265: 264: 259: 257: 256: 241: 240: 217: 215: 214: 209: 207: 206: 191: 190: 124: 7296: 7295: 7289: 7288: 7287: 7285: 7284: 7283: 7279:Axiom of choice 7269: 7268: 7267: 7262: 7251: 7244: 7189:Category theory 7179:Algebraic logic 7162: 7133:Lambda calculus 7071:Church encoding 7057: 7033:Truth predicate 6889: 6855:Complete theory 6778: 6647: 6643: 6639: 6634: 6626: 6346: and 6342: 6337: 6323: 6299:New Foundations 6267:axiom of choice 6250: 6212:Gödel numbering 6152: and 6144: 6048: 5933: 5883: 5864: 5813:Boolean algebra 5799: 5763:Equiconsistency 5728:Classical logic 5705: 5686:Halting problem 5674: and 5650: and 5638: and 5637: 5632:Theorems ( 5627: 5544: 5539: 5509: 5504: 5431: 5410: 5394: 5359:New Foundations 5306: 5196: 5115:Cardinal number 5098: 5084: 5025: 4916: 4907: 4891: 4886: 4838:and Jean Rubin. 4836:Wayback Machine 4825:Wayback Machine 4802:Axiom of Choice 4798: 4770:. New edition. 4762:Translated in: 4714: 4685: 4667:Suppes, Patrick 4651: 4637:Schechter, Eric 4629: 4574: 4534:reprint, 2013, 4526: 4476: 4449: 4423: 4397: 4366: 4356:Springer-Verlag 4352:Axiom of Choice 4348:Herrlich, Horst 4326:Halmos, Paul R. 4302: 4297: 4296: 4263: 4259: 4251: 4247: 4237: 4235: 4227: 4226: 4222: 4215: 4211: 4202: 4201: 4197: 4189: 4185: 4148: 4144: 4135: 4134: 4130: 4107: 4103: 4093: 4089: 4081:It is shown by 4080: 4076: 4065: 4061: 4042: 4023:10.1.1.222.5699 3999: 3995: 3981: 3977: 3962: 3946:Category theory 3940: 3936: 3920:10.2307/1998165 3899: 3892: 3862: 3858: 3831: 3827: 3814: 3810: 3790: 3786: 3776: 3774: 3770: 3759: 3753: 3749: 3708:(12): 556–557. 3694: 3690: 3682: 3678: 3670: 3666: 3660: 3656: 3640: 3636: 3605:Martin-Löf, Per 3602: 3598: 3576: 3572: 3561: 3557: 3537: 3515: 3511: 3504: 3488: 3484: 3469: 3442: 3438: 3433: 3429: 3421: 3417: 3397: 3393: 3386: 3370: 3366: 3356:, p. 201: 3347: 3343: 3335: 3331: 3323: 3319: 3314: 3248: 3244: 3223: 3219: 3183: 3179: 3164: 3160: 3152: 3149: 3148: 3090: 3003: 2997: 2986: 2939: 2931:New Foundations 2886: 2869:endowed with a 2853: 2848: 2847: 2845: 2842: 2841: 2723: 2699: 2696: 2695: 2675: 2672: 2671: 2639: 2636: 2635: 2608:Tychonoff space 2524:are isomorphic. 2475:field extension 2392:Every infinite 2360: 2357: 2356: 2339: 2335: 2333: 2330: 2329: 2325:Every infinite 2322:are equivalent. 2302:natural numbers 2254: 2250: 2238: 2232: 2229: 2228: 2201: 2191: 2187: 2183: 2182: 2180: 2177: 2176: 2154: 2151: 2150: 2133: 2095: 2068: 2055: 2008:category theory 2004: 2002:Category theory 1989:connected graph 1858:Every set is a 1823:group structure 1796:Krull's theorem 1732:initial ordinal 1718:König's theorem 1637: 1594: 1592:Stronger axioms 1547:connected graph 1515:symmetric model 1479: 1472: 1465: 1458: 1454: 1447: 1440: 1433: 1426: 1419: 1409: 1408:. Then the set 1403: 1394: 1380: 1378: 1369: 1343: 1337: 1270: 1228:category theory 1199: 1071: 1027: 1006: 971:There is a set 828: 772: 749: 725:of the sets in 668: 664: 656: 653: 652: 608: 604: 593: 590: 589: 509: 475: 471: 463: 460: 459: 453: 432: 395:choice function 391: 348:choice function 313: 310: 309: 292: 288: 279: 275: 273: 270: 269: 246: 242: 236: 232: 227: 224: 223: 196: 192: 186: 182: 177: 174: 173: 139:axiom of choice 120: 118: 112: 103: 90: 73: 59: 49: 40: 24: 17: 12: 11: 5: 7294: 7293: 7282: 7281: 7264: 7263: 7249: 7246: 7245: 7243: 7242: 7237: 7232: 7227: 7222: 7221: 7220: 7210: 7205: 7200: 7191: 7186: 7181: 7176: 7174:Abstract logic 7170: 7168: 7164: 7163: 7161: 7160: 7155: 7153:Turing machine 7150: 7145: 7140: 7135: 7130: 7125: 7124: 7123: 7118: 7113: 7108: 7103: 7093: 7091:Computable set 7088: 7083: 7078: 7073: 7067: 7065: 7059: 7058: 7056: 7055: 7050: 7045: 7040: 7035: 7030: 7025: 7020: 7019: 7018: 7013: 7008: 6998: 6993: 6988: 6986:Satisfiability 6983: 6978: 6973: 6972: 6971: 6961: 6960: 6959: 6949: 6948: 6947: 6942: 6937: 6932: 6927: 6917: 6916: 6915: 6910: 6903:Interpretation 6899: 6897: 6891: 6890: 6888: 6887: 6882: 6877: 6872: 6867: 6857: 6852: 6851: 6850: 6849: 6848: 6838: 6833: 6823: 6818: 6813: 6808: 6803: 6798: 6792: 6790: 6784: 6783: 6780: 6779: 6777: 6776: 6768: 6767: 6766: 6765: 6760: 6759: 6758: 6753: 6748: 6728: 6727: 6726: 6724:minimal axioms 6721: 6710: 6709: 6708: 6697: 6696: 6695: 6690: 6685: 6680: 6675: 6670: 6657: 6655: 6636: 6635: 6633: 6632: 6631: 6630: 6618: 6613: 6612: 6611: 6606: 6601: 6596: 6586: 6581: 6576: 6571: 6570: 6569: 6564: 6554: 6553: 6552: 6547: 6542: 6537: 6527: 6522: 6521: 6520: 6515: 6510: 6500: 6499: 6498: 6493: 6488: 6483: 6478: 6473: 6463: 6458: 6453: 6448: 6447: 6446: 6441: 6436: 6431: 6421: 6416: 6414:Formation rule 6411: 6406: 6405: 6404: 6399: 6389: 6388: 6387: 6377: 6372: 6367: 6362: 6356: 6350: 6333:Formal systems 6329: 6328: 6325: 6324: 6322: 6321: 6316: 6311: 6306: 6301: 6296: 6291: 6286: 6281: 6276: 6275: 6274: 6269: 6258: 6256: 6252: 6251: 6249: 6248: 6247: 6246: 6236: 6231: 6230: 6229: 6222:Large cardinal 6219: 6214: 6209: 6204: 6199: 6185: 6184: 6183: 6178: 6173: 6158: 6156: 6146: 6145: 6143: 6142: 6141: 6140: 6135: 6130: 6120: 6115: 6110: 6105: 6100: 6095: 6090: 6085: 6080: 6075: 6070: 6065: 6059: 6057: 6050: 6049: 6047: 6046: 6045: 6044: 6039: 6034: 6029: 6024: 6019: 6011: 6010: 6009: 6004: 5994: 5989: 5987:Extensionality 5984: 5982:Ordinal number 5979: 5969: 5964: 5963: 5962: 5951: 5945: 5939: 5938: 5935: 5934: 5932: 5931: 5926: 5921: 5916: 5911: 5906: 5901: 5900: 5899: 5889: 5888: 5887: 5874: 5872: 5866: 5865: 5863: 5862: 5861: 5860: 5855: 5850: 5840: 5835: 5830: 5825: 5820: 5815: 5809: 5807: 5801: 5800: 5798: 5797: 5792: 5787: 5782: 5777: 5772: 5767: 5766: 5765: 5755: 5750: 5745: 5740: 5735: 5730: 5724: 5722: 5713: 5707: 5706: 5704: 5703: 5698: 5693: 5688: 5683: 5678: 5666:Cantor's 5664: 5659: 5654: 5644: 5642: 5629: 5628: 5626: 5625: 5620: 5615: 5610: 5605: 5600: 5595: 5590: 5585: 5580: 5575: 5570: 5565: 5564: 5563: 5552: 5550: 5546: 5545: 5538: 5537: 5530: 5523: 5515: 5506: 5505: 5503: 5502: 5497: 5495:Thoralf Skolem 5492: 5487: 5482: 5477: 5472: 5467: 5462: 5457: 5452: 5447: 5441: 5439: 5433: 5432: 5430: 5429: 5424: 5419: 5413: 5411: 5409: 5408: 5405: 5399: 5396: 5395: 5393: 5392: 5391: 5390: 5385: 5380: 5379: 5378: 5363: 5362: 5361: 5349: 5348: 5347: 5336: 5335: 5330: 5325: 5320: 5314: 5312: 5308: 5307: 5305: 5304: 5299: 5294: 5289: 5280: 5275: 5270: 5260: 5255: 5254: 5253: 5248: 5243: 5233: 5223: 5218: 5213: 5207: 5205: 5198: 5197: 5195: 5194: 5189: 5184: 5179: 5177:Ordinal number 5174: 5169: 5164: 5159: 5158: 5157: 5152: 5142: 5137: 5132: 5127: 5122: 5112: 5107: 5101: 5099: 5097: 5096: 5093: 5089: 5086: 5085: 5083: 5082: 5077: 5072: 5067: 5062: 5057: 5055:Disjoint union 5052: 5047: 5041: 5035: 5033: 5027: 5026: 5024: 5023: 5022: 5021: 5016: 5005: 5004: 5002:Martin's axiom 4999: 4994: 4989: 4984: 4979: 4974: 4969: 4967:Extensionality 4964: 4959: 4954: 4953: 4952: 4947: 4942: 4932: 4926: 4924: 4918: 4917: 4910: 4908: 4906: 4905: 4899: 4897: 4893: 4892: 4885: 4884: 4877: 4870: 4862: 4856: 4855: 4846:John Lane Bell 4839: 4815: 4809: 4797: 4796:External links 4794: 4793: 4792: 4791: 4790: 4789: 4788: 4785: 4757: 4756: 4743: 4707: 4689: 4683: 4663: 4649: 4633: 4627: 4611: 4600: 4591:, April 1970. 4578: 4572: 4559: 4543: 4524: 4507: 4495: 4481: 4439: 4427: 4421: 4401: 4395: 4375:Rubin, Jean E. 4373:Howard, Paul; 4370: 4364: 4344: 4322: 4301: 4298: 4295: 4294: 4275:(2): 149–163. 4257: 4245: 4220: 4209: 4195: 4191:Schechter 1996 4183: 4162:(3): 306–308. 4142: 4128: 4117:(1–2): 1–184. 4101: 4087: 4074: 4059: 4040: 3993: 3975: 3961:978-0199237180 3960: 3934: 3890: 3856: 3825: 3808: 3784: 3747: 3688: 3684:Mostowski 1938 3676: 3664: 3654: 3634: 3615:(3): 345–350. 3596: 3578:Per Martin-Löf 3570: 3555: 3535: 3509: 3502: 3482: 3467: 3436: 3427: 3425:, p. 240. 3415: 3391: 3384: 3364: 3362: 3361: 3360:mathematician. 3354:Mendelson 1964 3341: 3329: 3316: 3315: 3313: 3310: 3298: 3297: 3286: 3283: 3280: 3277: 3274: 3271: 3268: 3265: 3262: 3259: 3256: 3251: 3247: 3243: 3240: 3237: 3232: 3229: 3226: 3222: 3218: 3215: 3212: 3209: 3206: 3203: 3200: 3197: 3194: 3191: 3186: 3182: 3178: 3175: 3172: 3167: 3163: 3159: 3156: 3089: 3086: 3067: 3066: 3065:does not hold. 3059: 3040: 3033: 3026: 3023: 3020: 3017: 3009: 3006: 3001: 2993: 2984: 2955: 2938: 2935: 2905:large cardinal 2885: 2882: 2881: 2880: 2879: 2878: 2856: 2851: 2826: 2825: 2824: 2821: 2786: 2779: 2751:, there is an 2722: 2719: 2718: 2717: 2716: 2715: 2703: 2679: 2643: 2628: 2617: 2616: 2615: 2604: 2596: 2595: 2594: 2587: 2565: 2554: 2547: 2527: 2526: 2525: 2510: 2503: 2493: 2482: 2471: 2455: 2454: 2453: 2446: 2439: 2432: 2417:Vitali theorem 2411:Measure theory 2408: 2390: 2389: 2388: 2364: 2342: 2338: 2323: 2316: 2286: 2279:countable sets 2271: 2257: 2253: 2247: 2244: 2241: 2237: 2226: 2210: 2207: 2204: 2199: 2194: 2190: 2186: 2164: 2161: 2158: 2132: 2129: 2093: 2090:Hartogs number 2066: 2054: 2051: 2050: 2049: 2042: 2035: 2013:small category 2003: 2000: 1999: 1998: 1997: 1996: 1980: 1979: 1978: 1965:is the set of 1930: 1929: 1928: 1921: 1911: 1892: 1891: 1890: 1874: 1873: 1872: 1856: 1845: 1834: 1807: 1793: 1773: 1772: 1771: 1761: 1751: 1745: 1735: 1721: 1715: 1704: 1693: 1690: 1684: 1636: 1633: 1627: 1593: 1590: 1559:compact spaces 1477: 1470: 1463: 1452: 1445: 1438: 1431: 1424: 1417: 1399: 1390: 1374: 1365: 1353:introduced by 1336: 1333: 1305:extensionality 1269: 1266: 1198: 1195: 1070: 1067: 1026: 1023: 1005: 1002: 1001: 1000: 983:), there is a 965: 964: 929: 928: 905: 904: 863: 862: 851: 844: 838: 833:Given any set 825: 824: 821: 818: 815: 808: 807: 782:Given any set 771: 768: 767: 766: 763: 756: 748: 745: 744: 743: 702: 698: 695: 692: 689: 686: 683: 680: 677: 674: 671: 667: 663: 660: 639: 635: 632: 629: 626: 623: 620: 617: 614: 611: 607: 603: 600: 597: 582: 581: 570: 565: 561: 558: 555: 552: 549: 546: 543: 540: 536: 533: 530: 527: 523: 518: 515: 512: 508: 504: 501: 498: 495: 492: 488: 484: 481: 478: 474: 470: 467: 427: 390: 387: 323: 320: 317: 295: 291: 287: 282: 278: 255: 252: 249: 245: 239: 235: 231: 205: 202: 199: 195: 189: 185: 181: 171:indexed family 141:, abbreviated 114: 108: 99: 86: 76:indexed family 69: 55: 45: 36: 15: 9: 6: 4: 3: 2: 7292: 7291: 7280: 7277: 7276: 7274: 7261: 7260: 7255: 7247: 7241: 7238: 7236: 7233: 7231: 7228: 7226: 7223: 7219: 7216: 7215: 7214: 7211: 7209: 7206: 7204: 7201: 7199: 7195: 7192: 7190: 7187: 7185: 7182: 7180: 7177: 7175: 7172: 7171: 7169: 7165: 7159: 7156: 7154: 7151: 7149: 7148:Recursive set 7146: 7144: 7141: 7139: 7136: 7134: 7131: 7129: 7126: 7122: 7119: 7117: 7114: 7112: 7109: 7107: 7104: 7102: 7099: 7098: 7097: 7094: 7092: 7089: 7087: 7084: 7082: 7079: 7077: 7074: 7072: 7069: 7068: 7066: 7064: 7060: 7054: 7051: 7049: 7046: 7044: 7041: 7039: 7036: 7034: 7031: 7029: 7026: 7024: 7021: 7017: 7014: 7012: 7009: 7007: 7004: 7003: 7002: 6999: 6997: 6994: 6992: 6989: 6987: 6984: 6982: 6979: 6977: 6974: 6970: 6967: 6966: 6965: 6962: 6958: 6957:of arithmetic 6955: 6954: 6953: 6950: 6946: 6943: 6941: 6938: 6936: 6933: 6931: 6928: 6926: 6923: 6922: 6921: 6918: 6914: 6911: 6909: 6906: 6905: 6904: 6901: 6900: 6898: 6896: 6892: 6886: 6883: 6881: 6878: 6876: 6873: 6871: 6868: 6865: 6864:from ZFC 6861: 6858: 6856: 6853: 6847: 6844: 6843: 6842: 6839: 6837: 6834: 6832: 6829: 6828: 6827: 6824: 6822: 6819: 6817: 6814: 6812: 6809: 6807: 6804: 6802: 6799: 6797: 6794: 6793: 6791: 6789: 6785: 6775: 6774: 6770: 6769: 6764: 6763:non-Euclidean 6761: 6757: 6754: 6752: 6749: 6747: 6746: 6742: 6741: 6739: 6736: 6735: 6733: 6729: 6725: 6722: 6720: 6717: 6716: 6715: 6711: 6707: 6704: 6703: 6702: 6698: 6694: 6691: 6689: 6686: 6684: 6681: 6679: 6676: 6674: 6671: 6669: 6666: 6665: 6663: 6659: 6658: 6656: 6651: 6645: 6640:Example 6637: 6629: 6624: 6623: 6622: 6619: 6617: 6614: 6610: 6607: 6605: 6602: 6600: 6597: 6595: 6592: 6591: 6590: 6587: 6585: 6582: 6580: 6577: 6575: 6572: 6568: 6565: 6563: 6560: 6559: 6558: 6555: 6551: 6548: 6546: 6543: 6541: 6538: 6536: 6533: 6532: 6531: 6528: 6526: 6523: 6519: 6516: 6514: 6511: 6509: 6506: 6505: 6504: 6501: 6497: 6494: 6492: 6489: 6487: 6484: 6482: 6479: 6477: 6474: 6472: 6469: 6468: 6467: 6464: 6462: 6459: 6457: 6454: 6452: 6449: 6445: 6442: 6440: 6437: 6435: 6432: 6430: 6427: 6426: 6425: 6422: 6420: 6417: 6415: 6412: 6410: 6407: 6403: 6400: 6398: 6397:by definition 6395: 6394: 6393: 6390: 6386: 6383: 6382: 6381: 6378: 6376: 6373: 6371: 6368: 6366: 6363: 6361: 6358: 6357: 6354: 6351: 6349: 6345: 6340: 6334: 6330: 6320: 6317: 6315: 6312: 6310: 6307: 6305: 6302: 6300: 6297: 6295: 6292: 6290: 6287: 6285: 6284:Kripke–Platek 6282: 6280: 6277: 6273: 6270: 6268: 6265: 6264: 6263: 6260: 6259: 6257: 6253: 6245: 6242: 6241: 6240: 6237: 6235: 6232: 6228: 6225: 6224: 6223: 6220: 6218: 6215: 6213: 6210: 6208: 6205: 6203: 6200: 6197: 6193: 6189: 6186: 6182: 6179: 6177: 6174: 6172: 6169: 6168: 6167: 6163: 6160: 6159: 6157: 6155: 6151: 6147: 6139: 6136: 6134: 6131: 6129: 6128:constructible 6126: 6125: 6124: 6121: 6119: 6116: 6114: 6111: 6109: 6106: 6104: 6101: 6099: 6096: 6094: 6091: 6089: 6086: 6084: 6081: 6079: 6076: 6074: 6071: 6069: 6066: 6064: 6061: 6060: 6058: 6056: 6051: 6043: 6040: 6038: 6035: 6033: 6030: 6028: 6025: 6023: 6020: 6018: 6015: 6014: 6012: 6008: 6005: 6003: 6000: 5999: 5998: 5995: 5993: 5990: 5988: 5985: 5983: 5980: 5978: 5974: 5970: 5968: 5965: 5961: 5958: 5957: 5956: 5953: 5952: 5949: 5946: 5944: 5940: 5930: 5927: 5925: 5922: 5920: 5917: 5915: 5912: 5910: 5907: 5905: 5902: 5898: 5895: 5894: 5893: 5890: 5886: 5881: 5880: 5879: 5876: 5875: 5873: 5871: 5867: 5859: 5856: 5854: 5851: 5849: 5846: 5845: 5844: 5841: 5839: 5836: 5834: 5831: 5829: 5826: 5824: 5821: 5819: 5816: 5814: 5811: 5810: 5808: 5806: 5805:Propositional 5802: 5796: 5793: 5791: 5788: 5786: 5783: 5781: 5778: 5776: 5773: 5771: 5768: 5764: 5761: 5760: 5759: 5756: 5754: 5751: 5749: 5746: 5744: 5741: 5739: 5736: 5734: 5733:Logical truth 5731: 5729: 5726: 5725: 5723: 5721: 5717: 5714: 5712: 5708: 5702: 5699: 5697: 5694: 5692: 5689: 5687: 5684: 5682: 5679: 5677: 5673: 5669: 5665: 5663: 5660: 5658: 5655: 5653: 5649: 5646: 5645: 5643: 5641: 5635: 5630: 5624: 5621: 5619: 5616: 5614: 5611: 5609: 5606: 5604: 5601: 5599: 5596: 5594: 5591: 5589: 5586: 5584: 5581: 5579: 5576: 5574: 5571: 5569: 5566: 5562: 5559: 5558: 5557: 5554: 5553: 5551: 5547: 5543: 5536: 5531: 5529: 5524: 5522: 5517: 5516: 5513: 5501: 5500:Ernst Zermelo 5498: 5496: 5493: 5491: 5488: 5486: 5485:Willard Quine 5483: 5481: 5478: 5476: 5473: 5471: 5468: 5466: 5463: 5461: 5458: 5456: 5453: 5451: 5448: 5446: 5443: 5442: 5440: 5438: 5437:Set theorists 5434: 5428: 5425: 5423: 5420: 5418: 5415: 5414: 5412: 5406: 5404: 5401: 5400: 5397: 5389: 5386: 5384: 5383:Kripke–Platek 5381: 5377: 5374: 5373: 5372: 5369: 5368: 5367: 5364: 5360: 5357: 5356: 5355: 5354: 5350: 5346: 5343: 5342: 5341: 5338: 5337: 5334: 5331: 5329: 5326: 5324: 5321: 5319: 5316: 5315: 5313: 5309: 5303: 5300: 5298: 5295: 5293: 5290: 5288: 5286: 5281: 5279: 5276: 5274: 5271: 5268: 5264: 5261: 5259: 5256: 5252: 5249: 5247: 5244: 5242: 5239: 5238: 5237: 5234: 5231: 5227: 5224: 5222: 5219: 5217: 5214: 5212: 5209: 5208: 5206: 5203: 5199: 5193: 5190: 5188: 5185: 5183: 5180: 5178: 5175: 5173: 5170: 5168: 5165: 5163: 5160: 5156: 5153: 5151: 5148: 5147: 5146: 5143: 5141: 5138: 5136: 5133: 5131: 5128: 5126: 5123: 5120: 5116: 5113: 5111: 5108: 5106: 5103: 5102: 5100: 5094: 5091: 5090: 5087: 5081: 5078: 5076: 5073: 5071: 5068: 5066: 5063: 5061: 5058: 5056: 5053: 5051: 5048: 5045: 5042: 5040: 5037: 5036: 5034: 5032: 5028: 5020: 5019:specification 5017: 5015: 5012: 5011: 5010: 5007: 5006: 5003: 5000: 4998: 4995: 4993: 4990: 4988: 4985: 4983: 4980: 4978: 4975: 4973: 4970: 4968: 4965: 4963: 4960: 4958: 4955: 4951: 4948: 4946: 4943: 4941: 4938: 4937: 4936: 4933: 4931: 4928: 4927: 4925: 4923: 4919: 4914: 4904: 4901: 4900: 4898: 4894: 4890: 4883: 4878: 4876: 4871: 4869: 4864: 4863: 4860: 4853: 4852: 4847: 4843: 4840: 4837: 4833: 4830: 4826: 4822: 4819: 4816: 4813: 4810: 4807: 4803: 4800: 4799: 4786: 4783: 4782: 4781: 4780:0-674-32449-8 4777: 4773: 4769: 4765: 4761: 4760: 4759: 4758: 4755: 4751: 4747: 4746:Ernst Zermelo 4744: 4740: 4736: 4732: 4728: 4725:(4): 514–16. 4724: 4720: 4713: 4708: 4706: 4705:0-511-06659-7 4702: 4698: 4694: 4690: 4686: 4680: 4675: 4674: 4668: 4664: 4660: 4656: 4652: 4646: 4642: 4638: 4634: 4630: 4624: 4620: 4616: 4612: 4609: 4608:0-444-87708-8 4605: 4601: 4598: 4597:0-7204-2225-6 4594: 4590: 4586: 4585:Jean E. Rubin 4582: 4579: 4575: 4569: 4565: 4560: 4556: 4552: 4548: 4544: 4541: 4540:0-486-48841-1 4537: 4533: 4527: 4521: 4517: 4513: 4508: 4504: 4500: 4496: 4494: 4493:1-4020-8925-2 4490: 4486: 4482: 4475: 4470: 4465: 4461: 4457: 4456: 4448: 4444: 4440: 4436: 4432: 4428: 4424: 4418: 4414: 4410: 4406: 4402: 4398: 4396:9780821809778 4392: 4388: 4383: 4382: 4376: 4371: 4367: 4361: 4357: 4353: 4349: 4345: 4341: 4337: 4333: 4332: 4327: 4323: 4320: 4316: 4312: 4308: 4304: 4303: 4290: 4286: 4282: 4278: 4274: 4270: 4269: 4261: 4254: 4249: 4234: 4230: 4224: 4218: 4213: 4205: 4199: 4192: 4187: 4179: 4175: 4170: 4165: 4161: 4157: 4153: 4146: 4138: 4132: 4124: 4120: 4116: 4112: 4105: 4097: 4091: 4084: 4078: 4070: 4063: 4056: 4051: 4047: 4043: 4037: 4033: 4029: 4024: 4019: 4015: 4007: 4003: 3997: 3991: 3989: 3984: 3979: 3971: 3967: 3963: 3957: 3953: 3948: 3947: 3938: 3930: 3926: 3921: 3916: 3912: 3908: 3904: 3897: 3895: 3887: 3886:North-Holland 3883: 3882: 3877: 3873: 3869: 3865: 3860: 3852: 3848: 3844: 3840: 3836: 3829: 3822: 3818: 3812: 3805: 3801: 3798: 3794: 3788: 3769: 3765: 3758: 3751: 3743: 3739: 3734: 3729: 3724: 3719: 3715: 3711: 3707: 3703: 3699: 3692: 3685: 3680: 3673: 3672:Fraenkel 1922 3668: 3658: 3651: 3647: 3643: 3642:Errett Bishop 3638: 3630: 3626: 3622: 3618: 3614: 3610: 3606: 3600: 3593: 3589: 3585: 3584: 3579: 3574: 3568: 3566: 3559: 3551: 3546: 3542: 3538: 3532: 3528: 3524: 3520: 3513: 3505: 3503:9780486446172 3499: 3495: 3494: 3486: 3478: 3474: 3470: 3468:9780080887050 3464: 3460: 3459: 3454: 3450: 3446: 3440: 3431: 3424: 3419: 3412: 3408: 3404: 3400: 3399:Herrlich 2006 3395: 3387: 3385:9781000516333 3381: 3378:. CRC Press. 3377: 3376: 3368: 3358: 3357: 3355: 3351: 3345: 3339:, p. 351 3338: 3333: 3326: 3321: 3317: 3309: 3307: 3303: 3284: 3275: 3269: 3266: 3263: 3257: 3249: 3245: 3230: 3224: 3220: 3204: 3201: 3198: 3192: 3184: 3180: 3165: 3161: 3147: 3146: 3145: 3143: 3139: 3135: 3131: 3127: 3123: 3119: 3115: 3111: 3107: 3103: 3099: 3095: 3085: 3083: 3079: 3074: 3072: 3064: 3060: 3057: 3056:real analysis 3053: 3049: 3045: 3041: 3038: 3034: 3031: 3027: 3024: 3021: 3018: 3015: 3010: 3007: 3004: 2996: 2991: 2987: 2980: 2976: 2972: 2968: 2964: 2960: 2956: 2953: 2949: 2945: 2944: 2943: 2934: 2932: 2928: 2924: 2922: 2918: 2914: 2910: 2906: 2902: 2898: 2893: 2891: 2876: 2872: 2839: 2835: 2832: 2831: 2830: 2827: 2822: 2819: 2815: 2811: 2807: 2803: 2799: 2795: 2791: 2787: 2784: 2780: 2777: 2773: 2769: 2766: 2762: 2758: 2754: 2750: 2746: 2742: 2738: 2737: 2736: 2733: 2732: 2731: 2729: 2693: 2669: 2665: 2661: 2657: 2633: 2629: 2626: 2623: 2622: 2621: 2618: 2613: 2609: 2605: 2602: 2601: 2600: 2597: 2592: 2588: 2585: 2581: 2577: 2576:metric spaces 2574: 2570: 2566: 2563: 2559: 2555: 2552: 2551:Hilbert space 2548: 2545: 2541: 2537: 2533: 2532: 2531: 2528: 2523: 2522: 2517: 2516: 2511: 2508: 2504: 2501: 2497: 2494: 2491: 2487: 2483: 2480: 2476: 2472: 2469: 2465: 2461: 2460: 2459: 2456: 2451: 2447: 2444: 2440: 2437: 2433: 2430: 2426: 2422: 2418: 2414: 2413: 2412: 2409: 2406: 2402: 2398: 2395: 2391: 2386: 2382: 2378: 2362: 2340: 2336: 2328: 2324: 2321: 2317: 2314: 2310: 2306: 2303: 2299: 2295: 2291: 2287: 2284: 2280: 2276: 2272: 2270:is not empty. 2255: 2251: 2245: 2242: 2239: 2235: 2224: 2223:of non-empty 2208: 2205: 2202: 2197: 2192: 2188: 2184: 2159: 2156: 2148: 2144: 2143: 2142: 2139: 2138: 2137: 2128: 2126: 2122: 2118: 2114: 2110: 2106: 2101: 2099: 2091: 2087: 2083: 2079: 2074: 2072: 2064: 2060: 2047: 2043: 2040: 2036: 2033: 2029: 2025: 2024: 2023: 2020: 2018: 2014: 2009: 1994: 1993:spanning tree 1990: 1986: 1985: 1984: 1981: 1976: 1972: 1968: 1964: 1960: 1956: 1952: 1948: 1944: 1940: 1936: 1935: 1934: 1931: 1926: 1922: 1919: 1915: 1912: 1910:is connected. 1909: 1906: 1902: 1898: 1897: 1896: 1893: 1888: 1887:extreme point 1884: 1880: 1879: 1878: 1875: 1870: 1869: 1865: 1861: 1857: 1854: 1850: 1846: 1843: 1839: 1835: 1832: 1828: 1824: 1820: 1816: 1812: 1808: 1805: 1804:maximal ideal 1801: 1797: 1794: 1791: 1787: 1783: 1779: 1778: 1777: 1774: 1769: 1765: 1762: 1759: 1755: 1754:Tukey's lemma 1752: 1749: 1746: 1743: 1739: 1736: 1733: 1729: 1725: 1722: 1719: 1716: 1713: 1709: 1705: 1702: 1701:right inverse 1698: 1694: 1691: 1688: 1685: 1682: 1678: 1674: 1670: 1669:bijective map 1667:, there is a 1666: 1662: 1659: 1658: 1657: 1654: 1653: 1652: 1650: 1646: 1642: 1632: 1630: 1625: 1623: 1619: 1615: 1611: 1607: 1603: 1599: 1589: 1587: 1582: 1580: 1576: 1572: 1567: 1565: 1560: 1556: 1552: 1551:spanning tree 1548: 1544: 1540: 1536: 1535:maximal ideal 1532: 1526: 1523: 1518: 1516: 1512: 1508: 1504: 1500: 1496: 1492: 1487: 1485: 1476: 1469: 1462: 1451: 1444: 1437: 1430: 1423: 1416: 1412: 1407: 1402: 1398: 1393: 1389: 1383: 1377: 1373: 1368: 1364: 1360: 1356: 1352: 1348: 1342: 1332: 1328: 1324: 1322: 1321:Errett Bishop 1318: 1314: 1308: 1306: 1302: 1298: 1293: 1291: 1287: 1283: 1279: 1275: 1265: 1262: 1257: 1255: 1249: 1247: 1242: 1240: 1236: 1231: 1229: 1225: 1219: 1217: 1213: 1208: 1207:well-ordering 1204: 1194: 1192: 1187: 1185: 1181: 1177: 1173: 1169: 1165: 1161: 1157: 1153: 1149: 1145: 1141: 1137: 1133: 1129: 1124: 1122: 1118: 1114: 1110: 1106: 1102: 1098: 1094: 1090: 1086: 1080: 1077: 1066: 1064: 1060: 1056: 1052: 1048: 1044: 1040: 1036: 1032: 1022: 1021:collections. 1019: 1015: 1011: 998: 994: 990: 986: 982: 978: 974: 970: 969: 968: 962: 958: 954: 950: 946: 942: 938: 934: 933: 932: 926: 925: 924: 922: 918: 914: 910: 902: 898: 894: 890: 889: 888: 886: 881: 879: 875: 871: 868: 860: 856: 852: 849: 845: 842: 839: 836: 832: 831: 830: 822: 819: 816: 813: 812: 811: 805: 801: 797: 793: 789: 785: 781: 780: 779: 776: 764: 761: 757: 754: 753: 752: 741: 740: 739: 737: 732: 728: 724: 720: 715: 700: 693: 687: 681: 675: 669: 665: 661: 637: 630: 624: 615: 609: 605: 601: 587: 568: 563: 556: 553: 547: 541: 534: 531: 528: 521: 516: 513: 510: 506: 499: 496: 493: 482: 479: 476: 472: 468: 458: 457: 456: 452: 450: 446: 442: 438: 426: 424: 420: 416: 412: 408: 404: 400: 396: 386: 384: 380: 376: 372: 368: 363: 360: 355: 351: 349: 343: 341: 337: 336:Ernst Zermelo 321: 318: 315: 293: 289: 285: 280: 276: 253: 250: 247: 237: 233: 221: 203: 200: 197: 187: 183: 172: 168: 164: 162: 156: 152: 148: 144: 140: 136: 128: 123: 117: 111: 107: 102: 98: 94: 89: 84: 81: 77: 72: 66: 58: 54: 48: 44: 39: 35: 30: 26: 22: 7250: 7048:Ultraproduct 6895:Model theory 6860:Independence 6796:Formal proof 6788:Proof theory 6771: 6744: 6701:real numbers 6673:second-order 6584:Substitution 6461:Metalanguage 6402:conservative 6375:Axiom schema 6319:Constructive 6289:Morse–Kelley 6266: 6255:Set theories 6234:Aleph number 6227:inaccessible 6133:Grothendieck 6017:intersection 5904:Higher-order 5892:Second-order 5838:Truth tables 5795:Venn diagram 5578:Formal proof 5450:Georg Cantor 5445:Paul Bernays 5376:Morse–Kelley 5351: 5284: 5283:Subset 5230:hereditarily 5192:Venn diagram 5150:ordered pair 5065:Intersection 5009:Axiom schema 4934: 4849: 4767: 4749: 4722: 4718: 4692: 4672: 4640: 4618: 4581:Herman Rubin 4563: 4554: 4550: 4511: 4502: 4484: 4459: 4453: 4443:Lévy, Azriel 4434: 4431:Jech, Thomas 4408: 4405:Jech, Thomas 4380: 4351: 4330: 4310: 4272: 4266: 4260: 4248: 4236:. Retrieved 4232: 4223: 4212: 4198: 4186: 4159: 4155: 4145: 4131: 4114: 4110: 4104: 4090: 4077: 4068: 4062: 4013: 4005: 3996: 3987: 3978: 3945: 3937: 3910: 3906: 3879: 3871: 3867: 3859: 3834: 3828: 3811: 3799: 3787: 3775:. Retrieved 3763: 3750: 3705: 3701: 3691: 3679: 3667: 3657: 3649: 3637: 3612: 3608: 3599: 3591: 3581: 3573: 3564: 3558: 3548: 3518: 3512: 3492: 3485: 3457: 3453:Lévy, Azriel 3439: 3430: 3418: 3407:Zermelo 1904 3394: 3374: 3367: 3349: 3344: 3332: 3325:Zermelo 1904 3320: 3305: 3302:axiom scheme 3299: 3141: 3137: 3133: 3129: 3125: 3121: 3117: 3113: 3109: 3105: 3101: 3097: 3091: 3075: 3068: 2999: 2994: 2989: 2982: 2978: 2970: 2966: 2962: 2958: 2951: 2940: 2925: 2894: 2887: 2874: 2837: 2817: 2813: 2809: 2805: 2801: 2797: 2775: 2771: 2767: 2760: 2756: 2748: 2744: 2727: 2724: 2714:has a model. 2654:is a set of 2519: 2513: 2486:vector space 2427:that is not 2425:real numbers 2404: 2400: 2399:satisfies 2× 2396: 2308: 2304: 2289: 2134: 2102: 2097: 2085: 2081: 2077: 2075: 2056: 2053:Weaker forms 2046:left-adjoint 2026:Every small 2021: 2005: 1983:Graph theory 1966: 1962: 1958: 1954: 1950: 1946: 1938: 1866: 1827:cancellative 1818: 1810: 1789: 1782:vector space 1741: 1738:Zorn's lemma 1711: 1680: 1676: 1672: 1664: 1645:Zorn's lemma 1638: 1595: 1583: 1568: 1553:, and every 1539:vector space 1527: 1519: 1494: 1488: 1483: 1474: 1467: 1460: 1449: 1442: 1435: 1428: 1421: 1414: 1410: 1405: 1400: 1396: 1391: 1387: 1381: 1375: 1371: 1366: 1362: 1344: 1335:Independence 1330: 1326: 1309: 1294: 1278:topos theory 1271: 1258: 1250: 1243: 1232: 1220: 1200: 1191:well-ordered 1188: 1179: 1175: 1171: 1167: 1163: 1159: 1155: 1151: 1147: 1143: 1139: 1135: 1131: 1127: 1125: 1120: 1116: 1112: 1108: 1100: 1096: 1092: 1089:real numbers 1084: 1081: 1075: 1072: 1058: 1054: 1050: 1046: 1042: 1038: 1034: 1030: 1028: 1017: 1007: 996: 992: 988: 984: 980: 976: 972: 966: 960: 956: 952: 948: 944: 940: 936: 935:For any set 930: 920: 916: 912: 908: 906: 900: 892: 891:For any set 884: 882: 877: 873: 869: 864: 858: 854: 847: 840: 834: 826: 809: 803: 799: 791: 787: 783: 777: 773: 750: 747:Nomenclature 735: 726: 718: 716: 583: 454: 448: 444: 440: 436: 435:For any set 428: 422: 418: 414: 410: 406: 402: 398: 392: 364: 358: 352: 344: 158: 146: 142: 138: 132: 126: 121: 115: 113:drawn from S 109: 105: 100: 96: 92: 87: 82: 80:real numbers 70: 56: 52: 46: 42: 37: 33: 25: 7158:Type theory 7106:undecidable 7038:Truth value 6925:equivalence 6604:non-logical 6217:Enumeration 6207:Isomorphism 6154:cardinality 6138:Von Neumann 6103:Ultrafilter 6068:Uncountable 6002:equivalence 5919:Quantifiers 5909:Fixed-point 5878:First-order 5758:Consistency 5743:Proposition 5720:Traditional 5691:Lindström's 5681:Compactness 5623:Type theory 5568:Cardinality 5475:Thomas Jech 5318:Alternative 5297:Uncountable 5251:Ultrafilter 5110:Cardinality 5014:replacement 4962:Determinacy 4829:Paul Howard 4313:: 253–257, 4238:10 November 3423:Suppes 1972 3411:Halmos 1960 3409:. See also 3403:Suppes 1972 3304:, in which 3144:of type σ: 3094:type theory 3071:Jech (2008) 2656:first-order 2562:compactness 2381:Baire space 2288:If the set 2125:ultrafilter 1817:defined on 1813:there is a 1635:Equivalents 1529:nontrivial 1499:inner model 1384:=1, 2, 3... 1134:by a group 135:mathematics 60:are finite. 6969:elementary 6662:arithmetic 6530:Quantifier 6508:functional 6380:Expression 6098:Transitive 6042:identities 6027:complement 5960:hereditary 5943:Set theory 5470:Kurt Gödel 5455:Paul Cohen 5292:Transitive 5060:Identities 5044:Complement 5031:Operations 4992:Regularity 4930:Adjunction 4889:Set theory 4844:entry by 4557:(8): 13–20 4340:0087.04403 4319:48.0199.02 4300:References 3817:Moore 2013 3044:measurable 2794:surjection 2741:surjection 2735:Set theory 2670:subset of 2660:zero-order 2498:needs the 2436:Borel sets 2385:determined 2379:subset of 2320:finite set 2141:Set theory 2039:equivalent 1842:projective 1687:Trichotomy 1656:Set theory 1507:Paul Cohen 1491:Kurt Gödel 1347:urelements 1339:See also: 1216:consistent 987:such that 959:) lies in 584:Thus, the 308:for every 268:such that 155:set theory 7240:Supertask 7143:Recursion 7101:decidable 6935:saturated 6913:of models 6836:deductive 6831:axiomatic 6751:Hilbert's 6738:Euclidean 6719:canonical 6642:axiomatic 6574:Signature 6503:Predicate 6392:Extension 6314:Ackermann 6239:Operation 6118:Universal 6108:Recursive 6083:Singleton 6078:Inhabited 6063:Countable 6053:Types of 6037:power set 6007:partition 5924:Predicate 5870:Predicate 5785:Syllogism 5775:Soundness 5748:Inference 5738:Tautology 5640:paradoxes 5403:Paradoxes 5323:Axiomatic 5302:Universal 5278:Singleton 5273:Recursive 5216:Countable 5211:Amorphous 5070:Power set 4987:Power set 4945:dependent 4940:countable 4739:124189935 4715:(reprint) 4669:(1972) . 4659:175294365 4617:(1993) . 4407:(2008) . 4289:119543439 4253:Jech 2008 4178:122334025 4096:Lévy 1958 4083:Jech 2008 4018:CiteSeerX 3970:740446073 3913:: 31–59. 3864:A. 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