hpprc/enwiki-paragraphs · Datasets at Hugging Face

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Austrian German	825	Standard Austrian German	For many years, Austria had a special form of the language for official government documents that is known as Österreichische Kanzleisprache, or "Austrian chancellery language". It is a very traditional form of the language, probably derived from medieval deeds and documents, and has a very complex structure and vocabulary generally reserved for such documents. For most speakers (even native speakers), this form of the language is generally difficult to understand, as it contains many highly-specialised terms for diplomatic, internal, official, and military matters. There are no regional variations because the special written form has been used mainly by a government that has now for centuries been based in Vienna.
Austrian German	825	Standard Austrian German	Österreichische Kanzleisprache is now used less and less because of various administrative reforms that reduced the number of traditional civil servants (Beamte). As a result, Standard Austrian German is replacing it in government and administrative texts.
Austrian German	825	Standard Austrian German	When Austria became a member of the European Union, 23 food-related terms were listed in its accession agreement as having the same legal status as the equivalent terms used in Germany, for example, the words for "potato", "tomato", and "Brussels sprouts". (Examples in "Vocabulary") Austrian German is the only variety of a pluricentric language recognized under international law or EU primary law.
Austrian German	825	Standard Austrian German	In Austria, as in the German-speaking parts of Switzerland and in southern Germany, verbs that express a state tend to use sein as the auxiliary verb in the perfect, as well as verbs of movement. Verbs which fall into this category include sitzen (to sit), liegen (to lie) and, in parts of Styria and Carinthia, schlafen (to sleep). Therefore, the perfect of these verbs would be ich bin gesessen, ich bin gelegen and ich bin geschlafen, respectively.
Austrian German	825	Standard Austrian German	In Germany, the words stehen (to stand) and gestehen (to confess) are identical in the present perfect: habe gestanden. The Austrian variant avoids that potential ambiguity (bin gestanden from stehen, "to stand"; and habe gestanden from gestehen, "to confess": "der Verbrecher ist vor dem Richter gestanden und hat gestanden").
Austrian German	825	Standard Austrian German	In addition, the preterite (simple past) is very rarely used in Austria, especially in the spoken language, with the exception of some modal verbs (ich sollte, ich wollte).
Austrian German	825	Standard Austrian German	There are many official terms that differ in Austrian German from their usage in most parts of Germany. Words used in Austria are Jänner (January) rather than Januar, Feber (seldom, February) along with Februar, heuer (this year) along with dieses Jahr, Stiege (stairs) along with Treppen, Rauchfang (chimney) instead of Schornstein, many administrative, legal and political terms, and many food terms, including the following:
Austrian German	825	Standard Austrian German
Austrian German	825	Standard Austrian German	There are, however, some false friends between the two regional varieties:
Austrian German	825	Dialects	In addition to the standard variety, in everyday life most Austrians speak one of a number of Upper German dialects.
Austrian German	825	Dialects	While strong forms of the various dialects are not fully mutually intelligible to northern Germans, communication is much easier in Bavaria, especially rural areas, where the Bavarian dialect still predominates as the mother tongue. The Central Austro-Bavarian dialects are more intelligible to speakers of Standard German than the Southern Austro-Bavarian dialects of Tyrol.
Austrian German	825	Dialects	Viennese, the Austro-Bavarian dialect of Vienna, is seen for many in Germany as quintessentially Austrian. The people of Graz, the capital of Styria, speak yet another dialect which is not very Styrian and more easily understood by people from other parts of Austria than other Styrian dialects, for example from western Styria.
Austrian German	825	Dialects	Simple words in the various dialects are very similar, but pronunciation is distinct for each and, after listening to a few spoken words, it may be possible for an Austrian to realise which dialect is being spoken. However, in regard to the dialects of the deeper valleys of the Tyrol, other Tyroleans are often unable to understand them. Speakers from the different provinces of Austria can easily be distinguished from each other by their particular accents (probably more so than Bavarians), those of Carinthia, Styria, Vienna, Upper Austria, and the Tyrol being very characteristic. Speakers from those regions, even those speaking Standard German, can usually be easily identified by their accent, even by an untrained listener.
Austrian German	825	Dialects	Several of the dialects have been influenced by contact with non-Germanic linguistic groups, such as the dialect of Carinthia, where, in the past, many speakers were bilingual (and, in the southeastern portions of the state, many still are even today) with Slovene, and the dialect of Vienna, which has been influenced by immigration during the Austro-Hungarian period, particularly from what is today the Czech Republic. The German dialects of South Tyrol have been influenced by local Romance languages, particularly noticeable with the many loanwords from Italian and Ladin.
Austrian German	825	Dialects	The geographic borderlines between the different accents (isoglosses) coincide strongly with the borders of the states and also with the border with Bavaria, with Bavarians having a markedly different rhythm of speech in spite of the linguistic similarities.
Axiom of choice	840		In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to construct a new set by choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family ( S i ) i ∈ I {\displaystyle (S_{i})_{i\in I}} of nonempty sets, there exists an indexed set ( x i ) i ∈ I {\displaystyle (x_{i})_{i\in I}} such that x i ∈ S i {\displaystyle x_{i}\in S_{i}} for every i ∈ I {\displaystyle i\in I} . The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.
Axiom of choice	840		In many cases, a set created by choosing elements can be made without invoking the axiom of choice; this is, in particular, if the number of sets from which to choose the elements is finite, or if a canonical rule on how to choose the elements is available – some distinguishing property that happens to hold for exactly one element in each set. An illustrative example is sets picked from the natural numbers. From such sets, one may always select the smallest number, e.g. given the sets {{4, 5, 6}, {10, 12}, {1, 400, 617, 8000}}, the set containing each smallest element is {4, 10, 1}. In this case, "select the smallest number" is a choice function. Even if infinitely many sets were 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. However, 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.
Axiom of choice	840		Bertrand Russell coined an analogy: for any (even infinite) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate collection (i.e. set) of shoes; this makes it possible to define a choice function directly. For an infinite collection of pairs of socks (assumed to have no distinguishing features), 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.
Axiom of choice	840		Although originally controversial, the axiom of choice is now used without reservation by most mathematicians, 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.
Axiom of choice	840	Statement	A choice function (also called selector or selection) is a function f, defined on a collection X of nonempty sets, such that for every set A in X, f(A) is an element of A. With this concept, the axiom can be stated:
Axiom of choice	840	Statement	Axiom — For any set X of nonempty sets, there exists a choice function f that is defined on X and maps each set of X to an element of that set.
Axiom of choice	840	Statement	Formally, this may be expressed as follows:
Axiom of choice	840	Statement	Thus, the negation of the axiom of choice states that there exists a collection of nonempty sets that has no choice function. ( p → q ⟺ ¬ [ p ∧ ( ¬ q ) ] {\displaystyle p\rightarrow q\Longleftrightarrow \lnot [p\land (\lnot q)]} , so ¬ ( p → q ) ⟺ p ∧ ( ¬ q ) {\displaystyle \lnot (p\rightarrow q)\Longleftrightarrow p\land (\lnot q)} where ¬ {\displaystyle \lnot } is negation.)
Axiom of choice	840	Statement	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:
Axiom of choice	840	Statement	In this article and other discussions of the Axiom of Choice the following abbreviations are common:
Axiom of choice	840	Statement	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.
Axiom of choice	840	Statement	One variation avoids the use of choice functions by, in effect, replacing each choice function with its range:
Axiom of choice	840	Statement	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.
Axiom of choice	840	Statement	Another equivalent axiom only considers collections X that are essentially powersets of other sets:
Axiom of choice	840	Statement	Authors who use this formulation often speak of the choice function on A, but this is a slightly different notion of choice function. Its domain is the power set of A (with the empty set removed), and so makes sense for any set A, whereas with the definition used elsewhere in this article, the domain of a choice function on a collection of sets is that collection, and so only makes sense for sets of sets. With this alternate notion of choice function, the axiom of choice can be compactly stated as
Axiom of choice	840	Statement	which is equivalent to
Axiom of choice	840	Statement	The negation of the axiom can thus be expressed as:
Axiom of choice	840	Statement	The usual statement of the axiom of choice does not specify whether the collection of nonempty sets is finite or infinite, and thus implies that every finite collection of nonempty sets has a choice function. However, that particular case is a theorem of the Zermelo–Fraenkel set theory without the axiom of choice (ZF); it is easily proved by the principle of finite induction. 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.
Axiom of choice	840	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" to define a function F. In general, it is impossible to prove that F exists without the axiom of choice, but this seems to have gone unnoticed until Zermelo.
Axiom of choice	840	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 add the axiom of choice to our axioms of set theory.
Axiom of choice	840	Examples	The difficulty appears when there is no natural choice of elements from each set. If we cannot make explicit choices, how do we know that our selection forms a legitimate set (as defined by the other ZF axioms of set theory)? For example, suppose that X is the set of all non-empty subsets of the real numbers. First we might try to proceed as if X were finite. If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently, we shall never be able to produce a choice function for all of X. Next we might try specifying the least element from each set. But some subsets of the real numbers do not have least elements. For example, the open interval (0,1) does not have a least element: if x is in (0,1), then so is x/2, and x/2 is always strictly smaller than x. So this attempt also fails.
Axiom of choice	840	Examples	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 form a set from selecting a point in each orbit requires that one add the axiom of choice to our axioms of set theory. See non-measurable set for more details.
Axiom of choice	840	Examples	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.
Axiom of choice	840	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.
Axiom of choice	840	Criticism and acceptance	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 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.
Axiom of choice	840	Criticism and acceptance	Another argument against the axiom of choice is that it implies the existence of objects that may seem counterintuitive. 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.
Axiom of choice	840	Criticism and acceptance	Moreover, paradoxical consequences of the axiom of choice for the no-signaling principle in physics have recently been pointed out.
Axiom of choice	840	Criticism and acceptance	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.
Axiom of choice	840	Criticism and acceptance	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 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.
Axiom of choice	840	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.
Axiom of choice	840	In 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. Errett Bishop argued that the axiom of choice was constructively acceptable, saying
Axiom of choice	840	In constructive mathematics	A choice function exists in constructive mathematics, because a choice is implied by the very meaning of existence.
Axiom of choice	840	In constructive mathematics	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.
Axiom of choice	840	In constructive mathematics	Some results in constructive set theory use the axiom of countable choice or the axiom of dependent choice, which do not imply the law of the excluded middle in constructive set theory. Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.
Axiom of choice	840	Independence	In 1938, 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 if ZF itself is consistent. In 1963, Paul Cohen employed the technique of forcing, 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 which satisfies ZF¬C (ZF with the negation of AC added as axiom) and thus showing that ZF¬C is consistent.
Axiom of choice	840	Independence	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 POSSIBLY 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.
Axiom of choice	840	Independence	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: the cardinalities of any two sets are comparable, every nontrivial ring with unity has a maximal ideal, every vector space has a basis, every connected graph has a spanning tree, and every product of compact spaces is compact, among many others. Without the axiom of choice, these theorems may not hold for mathematical objects of large cardinality.
Axiom of choice	840	Independence	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. 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.
Axiom of choice	840	Independence	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.
Axiom of choice	840	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 an axiom called the axiom of global choice that is stronger than the axiom of choice for sets because it also applies to proper classes. The axiom of global choice follows from the axiom of limitation of size. Tarski's axiom, which is used in Tarski–Grothendieck set theory and states (in the vernacular) that every set belongs to some Grothendieck universe, is stronger than the axiom of choice.
Axiom of choice	840	Equivalents	There are important statements that, assuming the axioms of ZF but neither AC nor ¬AC, are equivalent to the axiom of choice. 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.
Axiom of choice	840	Equivalents	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.
Axiom of choice	840	Equivalents	Examples of category-theoretic statements which require choice include:
Axiom of choice	840	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.
Axiom of choice	840	Weaker forms	Given an ordinal parameter α ≥ ω+2 — for every set S with rank less than α, S is well-orderable. Given an ordinal parameter α ≥ 1 — for every set S with Hartogs number less than ωα, S is well-orderable. As the ordinal parameter is increased, these approximate the full axiom of choice more and more closely.
Axiom of choice	840	Weaker forms	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 Tarski's 1930 ultrafilter lemma: every filter is a subset of some ultrafilter.
Axiom of choice	840	Weaker forms	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.
Axiom of choice	840	Weaker forms	There are several historically important set-theoretic statements implied by AC whose equivalence to AC is open. The partition principle, which was formulated before AC itself, was cited by Zermelo as a justification for believing AC. In 1906, Russell declared PP to be equivalent, but whether the partition principle implies AC is still the oldest open problem in set theory, and the equivalences of the other statements are similarly hard old open problems. In every known model of ZF where choice fails, these statements fail too, but it is unknown if they can hold without choice.
Axiom of choice	840	Stronger forms of the negation of AC	If we abbreviate by BP the claim that every set of real numbers has the property of Baire, then BP is stronger than ¬AC, which asserts the nonexistence of any choice function on perhaps only a single set of nonempty sets. Strengthened negations may be compatible with weakened forms of AC. For example, ZF + DC + BP is consistent, if ZF is.
Axiom of choice	840	Stronger forms of the negation of AC	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).
Axiom of choice	840	Stronger forms of the negation of AC	Quine's system of axiomatic set theory, New Foundations (NF), takes its name from the title ("New Foundations for Mathematical Logic") of the 1937 article which introduced it. In the NF axiomatic system, the axiom of choice can be disproved.
Axiom of choice	840	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. 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.
Axiom of choice	840	Statements consistent with the negation of AC	For proofs, see Jech (2008).
Axiom of choice	840	Statements consistent with the negation of AC	Additionally, by imposing definability conditions on sets (in the sense of descriptive set theory) one can often prove restricted versions of the axiom of choice from axioms incompatible with general choice. This appears, for example, in the Moschovakis coding lemma.
Axiom of choice	840	Axiom of choice in type theory	In type theory, a different kind of statement is known as the axiom of choice. This form begins with two types, σ and τ, and a relation R between objects of type σ and objects of type τ. The axiom of choice states that if for each x of type σ there exists a y of type τ such that R(x,y), then there is a function f from objects of type σ to objects of type τ such that R(x,f(x)) holds for all x of type σ:
Axiom of choice	840	Axiom of choice in type theory	Unlike in set theory, the axiom of choice in type theory is typically stated as an axiom scheme, in which R varies over all formulas or over all formulas of a particular logical form.
Axiom of choice	840	Quotations	The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?
Axiom of choice	840	Quotations	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.
Axiom of choice	840	Quotations	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.
Axiom of choice	840	Quotations	The observation here is that one can define a function to select from an infinite number of pairs of shoes, for example by choosing the left shoe from each pair. 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.
Axiom of choice	840	Quotations	Tarski tried to publish his theorem [the equivalence between AC and "every infinite set A has the same cardinality as A × A", 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.
Axiom of choice	840	Quotations	Polish-American mathematician Jan Mycielski relates this anecdote in a 2006 article in the Notices of the AMS.
Axiom of choice	840	Quotations	The axiom gets its name not because mathematicians prefer it to other axioms.
Axiom of choice	840	Quotations	This quote comes from the famous April Fools' Day article in the computer recreations column of the Scientific American, April 1989.
Attila	841		Attila (/əˈtɪlə/ ə-TIL-ə or /ˈætɪlə/ AT-il-ə; fl. c. 406–453), frequently called Attila the Hun, was the ruler of the Huns from 434 until his death, in March 453. He was also the leader of an empire consisting of Huns, Ostrogoths, Alans, and Bulgars, among others, in Central and Eastern Europe.
Attila	841		As nephews to Rugila, Attila and his elder brother Bleda succeeded him to the throne in 435, ruling jointly until the death of Bleda in 445. During his reign, Attila was one of the most feared enemies of the Western and Eastern Roman Empires. He crossed the Danube twice and plundered the Balkans but was unable to take Constantinople. In 441, he led an invasion of the Eastern Roman (Byzantine) Empire, the success of which emboldened him to invade the West. He also attempted to conquer Roman Gaul (modern France), crossing the Rhine in 451 and marching as far as Aurelianum (Orléans), before being stopped in the Battle of the Catalaunian Plains.
Attila	841		He subsequently invaded Italy, devastating the northern provinces, but was unable to take Rome. He planned for further campaigns against the Romans but died in 453. After Attila's death, his close adviser, Ardaric of the Gepids, led a Germanic revolt against Hunnic rule, after which the Hunnic Empire quickly collapsed. Attila lived on as a character in Germanic heroic legend.
Attila	841	Etymology	Many scholars have argued that the name Attila derives from East Germanic origin; Attila is formed from the Gothic or Gepidic noun atta, "father", by means of the diminutive suffix -ila, meaning "little father", compare Wulfila from wulfs "wolf" and -ila, i.e. "little wolf". The Gothic etymology was first proposed by Jacob and Wilhelm Grimm in the early 19th century. Maenchen-Helfen notes that this derivation of the name "offers neither phonetic nor semantic difficulties", and Gerhard Doerfer notes that the name is simply correct Gothic. Alexander Savelyev and Choongwon Jeong (2020) similarly state that Attila's name "must have been Gothic in origin." The name has sometimes been interpreted as a Germanization of a name of Hunnic origin.
Attila	841	Etymology	Other scholars have argued for a Turkic origin of the name. Omeljan Pritsak considered Ἀττίλα (Attíla) a composite title-name which derived from Turkic es (great, old), and til (sea, ocean), and the suffix /a/. The stressed back syllabic til assimilated the front member es, so it became as. It is a nominative, in form of attíl- (< etsíl < *es tíl) with the meaning "the oceanic, universal ruler". J. J. Mikkola connected it with Turkic āt (name, fame).
Attila	841	Etymology	As another Turkic possibility, H. Althof (1902) considered it was related to Turkish atli (horseman, cavalier), or Turkish at (horse) and dil (tongue). Maenchen-Helfen argues that Pritsak's derivation is "ingenious but for many reasons unacceptable", while dismissing Mikkola's as "too farfetched to be taken seriously". M. Snædal similarly notes that none of these proposals has achieved wide acceptance.
Attila	841	Etymology	Criticizing the proposals of finding Turkic or other etymologies for Attila, Doerfer notes that King George VI of the United Kingdom had a name of Greek origin, and Süleyman the Magnificent had a name of Arabic origin, yet that does not make them Greeks or Arabs: it is therefore plausible that Attila would have a name not of Hunnic origin. Historian Hyun Jin Kim, however, has argued that the Turkic etymology is "more probable".
Attila	841	Etymology	M. Snædal, in a paper that rejects the Germanic derivation but notes the problems with the existing proposed Turkic etymologies, argues that Attila's name could have originated from Turkic-Mongolian at, adyy/agta (gelding, warhorse) and Turkish atlı (horseman, cavalier), meaning "possessor of geldings, provider of warhorses".
Attila	841	Historiography and source	The historiography of Attila is faced with a major challenge, in that the only complete sources are written in Greek and Latin by the enemies of the Huns. Attila's contemporaries left many testimonials of his life, but only fragments of these remain. Priscus was a Byzantine diplomat and historian who wrote in Greek, and he was both a witness to and an actor in the story of Attila, as a member of the embassy of Theodosius II at the Hunnic court in 449. He was obviously biased by his political position, but his writing is a major source for information on the life of Attila, and he is the only person known to have recorded a physical description of him. He wrote a history of the late Roman Empire in eight books covering the period from 430 to 476.
Attila	841	Historiography and source	Only fragments of Priscus' work remain. It was cited extensively by 6th-century historians Procopius and Jordanes, especially in Jordanes' The Origin and Deeds of the Goths, which contains numerous references to Priscus's history, and it is also an important source of information about the Hunnic empire and its neighbors. He describes the legacy of Attila and the Hunnic people for a century after Attila's death. Marcellinus Comes, a chancellor of Justinian during the same era, also describes the relations between the Huns and the Eastern Roman Empire.
Attila	841	Historiography and source	Numerous ecclesiastical writings contain useful but scattered information, sometimes difficult to authenticate or distorted by years of hand-copying between the 6th and 17th centuries. The Hungarian writers of the 12th century wished to portray the Huns in a positive light as their glorious ancestors, and so repressed certain historical elements and added their own legends.
Attila	841	Historiography and source	The literature and knowledge of the Huns themselves was transmitted orally, by means of epics and chanted poems that were handed down from generation to generation. Indirectly, fragments of this oral history have reached us via the literature of the Scandinavians and Germans, neighbors of the Huns who wrote between the 9th and 13th centuries. Attila is a major character in many Medieval epics, such as the Nibelungenlied, as well as various Eddas and sagas.
Attila	841	Historiography and source	Archaeological investigation has uncovered some details about the lifestyle, art, and warfare of the Huns. There are a few traces of battles and sieges, but the tomb of Attila and the location of his capital have not yet been found.
Attila	841	Early life and background	The Huns were a group of Eurasian nomads, appearing from east of the Volga, who migrated further into Western Europe c. 370 and built up an enormous empire there. Their main military techniques were mounted archery and javelin throwing. They were in the process of developing settlements before their arrival in Western Europe, yet the Huns were a society of pastoral warriors whose primary form of nourishment was meat and milk, products of their herds.
Attila	841	Early life and background	The origin and language of the Huns has been the subject of debate for centuries. According to some theories, their leaders at least may have spoken a Turkic language, perhaps closest to the modern Chuvash language. According to the Encyclopedia of European Peoples, "the Huns, especially those who migrated to the west, may have been a combination of central Asian Turkic, Mongolic, and Ugric stocks".
Attila	841	Early life and background	Attila's father Mundzuk was the brother of kings Octar and Ruga, who reigned jointly over the Hunnic empire in the early fifth century. This form of diarchy was recurrent with the Huns, but historians are unsure whether it was institutionalized, merely customary, or an occasional occurrence. His family was from a noble lineage, but it is uncertain whether they constituted a royal dynasty. Attila's birthdate is debated; journalist Éric Deschodt and writer Herman Schreiber have proposed a date of 395. However, historian Iaroslav Lebedynsky and archaeologist Katalin Escher prefer an estimate between the 390s and the first decade of the fifth century. Several historians have proposed 406 as the date.
Attila	841	Early life and background	Attila grew up in a rapidly changing world. His people were nomads who had only recently arrived in Europe. They crossed the Volga river during the 370s and annexed the territory of the Alans, then attacked the Gothic kingdom between the Carpathian mountains and the Danube. They were a very mobile people, whose mounted archers had acquired a reputation for invincibility, and the Germanic tribes seemed unable to withstand them. Vast populations fleeing the Huns moved from Germania into the Roman Empire in the west and south, and along the banks of the Rhine and Danube. In 376, the Goths crossed the Danube, initially submitting to the Romans but soon rebelling against Emperor Valens, whom they killed in the Battle of Adrianople in 378. Large numbers of Vandals, Alans, Suebi, and Burgundians crossed the Rhine and invaded Roman Gaul on December 31, 406 to escape the Huns. The Roman Empire had been split in half since 395 and was ruled by two distinct governments, one based in Ravenna in the West, and the other in Constantinople in the East. The Roman Emperors, both East and West, were generally from the Theodosian family in Attila's lifetime (despite several power struggles).
Attila	841	Early life and background	The Huns dominated a vast territory with nebulous borders determined by the will of a constellation of ethnically varied peoples. Some were assimilated to Hunnic nationality, whereas many retained their own identities and rulers but acknowledged the suzerainty of the king of the Huns. The Huns were also the indirect source of many of the Romans' problems, driving various Germanic tribes into Roman territory, yet relations between the two empires were cordial: the Romans used the Huns as mercenaries against the Germans and even in their civil wars. Thus, the usurper Joannes was able to recruit thousands of Huns for his army against Valentinian III in 424. It was Aëtius, later Patrician of the West, who managed this operation. They exchanged ambassadors and hostages, the alliance lasting from 401 to 450 and permitting the Romans numerous military victories. The Huns considered the Romans to be paying them tribute, whereas the Romans preferred to view this as payment for services rendered. The Huns had become a great power by the time that Attila came of age during the reign of his uncle Ruga, to the point that Nestorius, the Patriarch of Constantinople, deplored the situation with these words: "They have become both masters and slaves of the Romans".
Attila	841	Campaigns against the Eastern Roman Empire	The death of Rugila (also known as Rua or Ruga) in 434 left the sons of his brother Mundzuk, Attila and Bleda, in control of the united Hun tribes. At the time of the two brothers' accession, the Hun tribes were bargaining with Eastern Roman Emperor Theodosius II's envoys for the return of several renegades who had taken refuge within the Eastern Roman Empire, possibly Hunnic nobles who disagreed with the brothers' assumption of leadership.
Attila	841	Campaigns against the Eastern Roman Empire	The following year, Attila and Bleda met with the imperial legation at Margus (Požarevac), all seated on horseback in the Hunnic manner, and negotiated an advantageous treaty. The Romans agreed to return the fugitives, to double their previous tribute of 350 Roman pounds (c. 115 kg) of gold, to open their markets to Hunnish traders, and to pay a ransom of eight solidi for each Roman taken prisoner by the Huns. The Huns, satisfied with the treaty, decamped from the Roman Empire and returned to their home in the Great Hungarian Plain, perhaps to consolidate and strengthen their empire. Theodosius used this opportunity to strengthen the walls of Constantinople, building the city's first sea wall, and to build up his border defenses along the Danube.
Attila	841	Campaigns against the Eastern Roman Empire	The Huns remained out of Roman sight for the next few years while they invaded the Sassanid Empire. They were defeated in Armenia by the Sassanids, abandoned their invasion, and turned their attentions back to Europe. In 440, they reappeared in force on the borders of the Roman Empire, attacking the merchants at the market on the north bank of the Danube that had been established by the treaty of 435.

Austrian German

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Standard Austrian German

For many years, Austria had a special form of the language for official government documents that is known as Österreichische Kanzleisprache, or "Austrian chancellery language". It is a very traditional form of the language, probably derived from medieval deeds and documents, and has a very complex structure and vocabulary generally reserved for such documents. For most speakers (even native speakers), this form of the language is generally difficult to understand, as it contains many highly-specialised terms for diplomatic, internal, official, and military matters. There are no regional variations because the special written form has been used mainly by a government that has now for centuries been based in Vienna.

Austrian German

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Standard Austrian German

Österreichische Kanzleisprache is now used less and less because of various administrative reforms that reduced the number of traditional civil servants (Beamte). As a result, Standard Austrian German is replacing it in government and administrative texts.

Austrian German

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Standard Austrian German

When Austria became a member of the European Union, 23 food-related terms were listed in its accession agreement as having the same legal status as the equivalent terms used in Germany, for example, the words for "potato", "tomato", and "Brussels sprouts". (Examples in "Vocabulary") Austrian German is the only variety of a pluricentric language recognized under international law or EU primary law.

Austrian German

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Standard Austrian German

In Austria, as in the German-speaking parts of Switzerland and in southern Germany, verbs that express a state tend to use sein as the auxiliary verb in the perfect, as well as verbs of movement. Verbs which fall into this category include sitzen (to sit), liegen (to lie) and, in parts of Styria and Carinthia, schlafen (to sleep). Therefore, the perfect of these verbs would be ich bin gesessen, ich bin gelegen and ich bin geschlafen, respectively.

Austrian German

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Standard Austrian German

In Germany, the words stehen (to stand) and gestehen (to confess) are identical in the present perfect: habe gestanden. The Austrian variant avoids that potential ambiguity (bin gestanden from stehen, "to stand"; and habe gestanden from gestehen, "to confess": "der Verbrecher ist vor dem Richter gestanden und hat gestanden").

Austrian German

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Standard Austrian German

In addition, the preterite (simple past) is very rarely used in Austria, especially in the spoken language, with the exception of some modal verbs (ich sollte, ich wollte).

Austrian German

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Standard Austrian German

There are many official terms that differ in Austrian German from their usage in most parts of Germany. Words used in Austria are Jänner (January) rather than Januar, Feber (seldom, February) along with Februar, heuer (this year) along with dieses Jahr, Stiege (stairs) along with Treppen, Rauchfang (chimney) instead of Schornstein, many administrative, legal and political terms, and many food terms, including the following:

Austrian German

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Standard Austrian German

Austrian German

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Standard Austrian German

There are, however, some false friends between the two regional varieties:

Austrian German

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Dialects

In addition to the standard variety, in everyday life most Austrians speak one of a number of Upper German dialects.

Austrian German

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Dialects

While strong forms of the various dialects are not fully mutually intelligible to northern Germans, communication is much easier in Bavaria, especially rural areas, where the Bavarian dialect still predominates as the mother tongue. The Central Austro-Bavarian dialects are more intelligible to speakers of Standard German than the Southern Austro-Bavarian dialects of Tyrol.

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Dialects

Viennese, the Austro-Bavarian dialect of Vienna, is seen for many in Germany as quintessentially Austrian. The people of Graz, the capital of Styria, speak yet another dialect which is not very Styrian and more easily understood by people from other parts of Austria than other Styrian dialects, for example from western Styria.

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Dialects

Simple words in the various dialects are very similar, but pronunciation is distinct for each and, after listening to a few spoken words, it may be possible for an Austrian to realise which dialect is being spoken. However, in regard to the dialects of the deeper valleys of the Tyrol, other Tyroleans are often unable to understand them. Speakers from the different provinces of Austria can easily be distinguished from each other by their particular accents (probably more so than Bavarians), those of Carinthia, Styria, Vienna, Upper Austria, and the Tyrol being very characteristic. Speakers from those regions, even those speaking Standard German, can usually be easily identified by their accent, even by an untrained listener.

Austrian German

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Dialects

Several of the dialects have been influenced by contact with non-Germanic linguistic groups, such as the dialect of Carinthia, where, in the past, many speakers were bilingual (and, in the southeastern portions of the state, many still are even today) with Slovene, and the dialect of Vienna, which has been influenced by immigration during the Austro-Hungarian period, particularly from what is today the Czech Republic. The German dialects of South Tyrol have been influenced by local Romance languages, particularly noticeable with the many loanwords from Italian and Ladin.

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Dialects

The geographic borderlines between the different accents (isoglosses) coincide strongly with the borders of the states and also with the border with Bavaria, with Bavarians having a markedly different rhythm of speech in spite of the linguistic similarities.

Axiom of choice

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In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to construct a new set by choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family ( S i ) i ∈ I {\displaystyle (S_{i})_{i\in I}} of nonempty sets, there exists an indexed set ( x i ) i ∈ I {\displaystyle (x_{i})_{i\in I}} such that x i ∈ S i {\displaystyle x_{i}\in S_{i}} for every i ∈ I {\displaystyle i\in I} . The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.

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In many cases, a set created by choosing elements can be made without invoking the axiom of choice; this is, in particular, if the number of sets from which to choose the elements is finite, or if a canonical rule on how to choose the elements is available – some distinguishing property that happens to hold for exactly one element in each set. An illustrative example is sets picked from the natural numbers. From such sets, one may always select the smallest number, e.g. given the sets {{4, 5, 6}, {10, 12}, {1, 400, 617, 8000}}, the set containing each smallest element is {4, 10, 1}. In this case, "select the smallest number" is a choice function. Even if infinitely many sets were 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. However, 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.

Axiom of choice

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Bertrand Russell coined an analogy: for any (even infinite) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate collection (i.e. set) of shoes; this makes it possible to define a choice function directly. For an infinite collection of pairs of socks (assumed to have no distinguishing features), 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.

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

Axiom of choice

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Statement

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

Axiom of choice

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Statement

Axiom — For any set X of nonempty sets, there exists a choice function f that is defined on X and maps each set of X to an element of that set.

Axiom of choice

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Statement

Formally, this may be expressed as follows:

Axiom of choice

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Statement

Thus, the negation of the axiom of choice states that there exists a collection of nonempty sets that has no choice function. ( p → q ⟺ ¬ [ p ∧ ( ¬ q ) ] {\displaystyle p\rightarrow q\Longleftrightarrow \lnot [p\land (\lnot q)]} , so ¬ ( p → q ) ⟺ p ∧ ( ¬ q ) {\displaystyle \lnot (p\rightarrow q)\Longleftrightarrow p\land (\lnot q)} where ¬ {\displaystyle \lnot } is negation.)

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Statement

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:

Axiom of choice

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Statement

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

Axiom of choice

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Statement

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.

Axiom of choice

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Statement

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

Axiom of choice

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Statement

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.

Axiom of choice

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Statement

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

Axiom of choice

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Statement

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

Axiom of choice

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Statement

which is equivalent to

Axiom of choice

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Statement

The negation of the axiom can thus be expressed as:

Axiom of choice

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Statement

The usual statement of the axiom of choice does not specify whether the collection of nonempty sets is finite or infinite, and thus implies that every finite collection of nonempty sets has a choice function. However, that particular case is a theorem of the Zermelo–Fraenkel set theory without the axiom of choice (ZF); it is easily proved by the principle of finite induction. 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.

Axiom of choice

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

Axiom of choice

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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 add the axiom of choice to our axioms of set theory.

Axiom of choice

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Examples

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

Axiom of choice

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Examples

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 form a set from selecting a point in each orbit requires that one add the axiom of choice to our axioms of set theory. See non-measurable set for more details.

Axiom of choice

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Examples

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.

Axiom of choice

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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.

Axiom of choice

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Criticism and acceptance

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 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.

Axiom of choice

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Criticism and acceptance

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

Axiom of choice

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Criticism and acceptance

Moreover, paradoxical consequences of the axiom of choice for the no-signaling principle in physics have recently been pointed out.

Axiom of choice

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Criticism and acceptance

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.

Axiom of choice

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Criticism and acceptance

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 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.

Axiom of choice

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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.

Axiom of choice

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In 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. Errett Bishop argued that the axiom of choice was constructively acceptable, saying

Axiom of choice

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In constructive mathematics

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

Axiom of choice

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In constructive mathematics

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.

Axiom of choice

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In constructive mathematics

Some results in constructive set theory use the axiom of countable choice or the axiom of dependent choice, which do not imply the law of the excluded middle in constructive set theory. Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.

Axiom of choice

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Independence

In 1938, 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 if ZF itself is consistent. In 1963, Paul Cohen employed the technique of forcing, 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 which satisfies ZF¬C (ZF with the negation of AC added as axiom) and thus showing that ZF¬C is consistent.

Axiom of choice

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Independence

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 POSSIBLY 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.

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Independence

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: the cardinalities of any two sets are comparable, every nontrivial ring with unity has a maximal ideal, every vector space has a basis, every connected graph has a spanning tree, and every product of compact spaces is compact, among many others. Without the axiom of choice, these theorems may not hold for mathematical objects of large cardinality.

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Independence

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. 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.

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Independence

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.

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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 an axiom called the axiom of global choice that is stronger than the axiom of choice for sets because it also applies to proper classes. The axiom of global choice follows from the axiom of limitation of size. Tarski's axiom, which is used in Tarski–Grothendieck set theory and states (in the vernacular) that every set belongs to some Grothendieck universe, is stronger than the axiom of choice.

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Equivalents

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

Axiom of choice

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Equivalents

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.

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Equivalents

Examples of category-theoretic statements which require choice include:

Axiom of choice

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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.

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Weaker forms

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

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Weaker forms

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 Tarski's 1930 ultrafilter lemma: every filter is a subset of some ultrafilter.

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Weaker forms

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.

Axiom of choice

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Weaker forms

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

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Stronger forms of the negation of AC

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

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Stronger forms of the negation of AC

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).

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Stronger forms of the negation of AC

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

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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. 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.

Axiom of choice

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Statements consistent with the negation of AC

For proofs, see Jech (2008).

Axiom of choice

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Statements consistent with the negation of AC

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

Axiom of choice

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Axiom of choice in type theory

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

Axiom of choice

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Axiom of choice in type theory

Unlike in set theory, the axiom of choice in type theory is typically stated as an axiom scheme, in which R varies over all formulas or over all formulas of a particular logical form.

Axiom of choice

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Quotations

The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?

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Quotations

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.

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Quotations

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.

Axiom of choice

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Quotations

The observation here is that one can define a function to select from an infinite number of pairs of shoes, for example by choosing the left shoe from each pair. 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.

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Quotations

Tarski tried to publish his theorem [the equivalence between AC and "every infinite set A has the same cardinality as A × A", 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.

Axiom of choice

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Quotations

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

Axiom of choice

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Quotations

The axiom gets its name not because mathematicians prefer it to other axioms.

Axiom of choice

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Quotations

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

Attila

841

Attila (/əˈtɪlə/ ə-TIL-ə or /ˈætɪlə/ AT-il-ə; fl. c. 406–453), frequently called Attila the Hun, was the ruler of the Huns from 434 until his death, in March 453. He was also the leader of an empire consisting of Huns, Ostrogoths, Alans, and Bulgars, among others, in Central and Eastern Europe.

Attila

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As nephews to Rugila, Attila and his elder brother Bleda succeeded him to the throne in 435, ruling jointly until the death of Bleda in 445. During his reign, Attila was one of the most feared enemies of the Western and Eastern Roman Empires. He crossed the Danube twice and plundered the Balkans but was unable to take Constantinople. In 441, he led an invasion of the Eastern Roman (Byzantine) Empire, the success of which emboldened him to invade the West. He also attempted to conquer Roman Gaul (modern France), crossing the Rhine in 451 and marching as far as Aurelianum (Orléans), before being stopped in the Battle of the Catalaunian Plains.

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He subsequently invaded Italy, devastating the northern provinces, but was unable to take Rome. He planned for further campaigns against the Romans but died in 453. After Attila's death, his close adviser, Ardaric of the Gepids, led a Germanic revolt against Hunnic rule, after which the Hunnic Empire quickly collapsed. Attila lived on as a character in Germanic heroic legend.

Attila

841

Etymology

Many scholars have argued that the name Attila derives from East Germanic origin; Attila is formed from the Gothic or Gepidic noun atta, "father", by means of the diminutive suffix -ila, meaning "little father", compare Wulfila from wulfs "wolf" and -ila, i.e. "little wolf". The Gothic etymology was first proposed by Jacob and Wilhelm Grimm in the early 19th century. Maenchen-Helfen notes that this derivation of the name "offers neither phonetic nor semantic difficulties", and Gerhard Doerfer notes that the name is simply correct Gothic. Alexander Savelyev and Choongwon Jeong (2020) similarly state that Attila's name "must have been Gothic in origin." The name has sometimes been interpreted as a Germanization of a name of Hunnic origin.

Attila

841

Etymology

Other scholars have argued for a Turkic origin of the name. Omeljan Pritsak considered Ἀττίλα (Attíla) a composite title-name which derived from Turkic *es (great, old), and *til (sea, ocean), and the suffix /a/. The stressed back syllabic til assimilated the front member es, so it became *as. It is a nominative, in form of attíl- (< *etsíl < *es tíl) with the meaning "the oceanic, universal ruler". J. J. Mikkola connected it with Turkic āt (name, fame).

Attila

841

Etymology

As another Turkic possibility, H. Althof (1902) considered it was related to Turkish atli (horseman, cavalier), or Turkish at (horse) and dil (tongue). Maenchen-Helfen argues that Pritsak's derivation is "ingenious but for many reasons unacceptable", while dismissing Mikkola's as "too farfetched to be taken seriously". M. Snædal similarly notes that none of these proposals has achieved wide acceptance.

Attila

841

Etymology

Criticizing the proposals of finding Turkic or other etymologies for Attila, Doerfer notes that King George VI of the United Kingdom had a name of Greek origin, and Süleyman the Magnificent had a name of Arabic origin, yet that does not make them Greeks or Arabs: it is therefore plausible that Attila would have a name not of Hunnic origin. Historian Hyun Jin Kim, however, has argued that the Turkic etymology is "more probable".

Attila

841

Etymology

M. Snædal, in a paper that rejects the Germanic derivation but notes the problems with the existing proposed Turkic etymologies, argues that Attila's name could have originated from Turkic-Mongolian at, adyy/agta (gelding, warhorse) and Turkish atlı (horseman, cavalier), meaning "possessor of geldings, provider of warhorses".

Attila

841

Historiography and source

The historiography of Attila is faced with a major challenge, in that the only complete sources are written in Greek and Latin by the enemies of the Huns. Attila's contemporaries left many testimonials of his life, but only fragments of these remain. Priscus was a Byzantine diplomat and historian who wrote in Greek, and he was both a witness to and an actor in the story of Attila, as a member of the embassy of Theodosius II at the Hunnic court in 449. He was obviously biased by his political position, but his writing is a major source for information on the life of Attila, and he is the only person known to have recorded a physical description of him. He wrote a history of the late Roman Empire in eight books covering the period from 430 to 476.

Attila

841

Historiography and source

Only fragments of Priscus' work remain. It was cited extensively by 6th-century historians Procopius and Jordanes, especially in Jordanes' The Origin and Deeds of the Goths, which contains numerous references to Priscus's history, and it is also an important source of information about the Hunnic empire and its neighbors. He describes the legacy of Attila and the Hunnic people for a century after Attila's death. Marcellinus Comes, a chancellor of Justinian during the same era, also describes the relations between the Huns and the Eastern Roman Empire.

Attila

841

Historiography and source

Numerous ecclesiastical writings contain useful but scattered information, sometimes difficult to authenticate or distorted by years of hand-copying between the 6th and 17th centuries. The Hungarian writers of the 12th century wished to portray the Huns in a positive light as their glorious ancestors, and so repressed certain historical elements and added their own legends.

Attila

841

Historiography and source

The literature and knowledge of the Huns themselves was transmitted orally, by means of epics and chanted poems that were handed down from generation to generation. Indirectly, fragments of this oral history have reached us via the literature of the Scandinavians and Germans, neighbors of the Huns who wrote between the 9th and 13th centuries. Attila is a major character in many Medieval epics, such as the Nibelungenlied, as well as various Eddas and sagas.

Attila

841

Historiography and source

Archaeological investigation has uncovered some details about the lifestyle, art, and warfare of the Huns. There are a few traces of battles and sieges, but the tomb of Attila and the location of his capital have not yet been found.

Attila

841

Early life and background

The Huns were a group of Eurasian nomads, appearing from east of the Volga, who migrated further into Western Europe c. 370 and built up an enormous empire there. Their main military techniques were mounted archery and javelin throwing. They were in the process of developing settlements before their arrival in Western Europe, yet the Huns were a society of pastoral warriors whose primary form of nourishment was meat and milk, products of their herds.

Attila

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Early life and background

The origin and language of the Huns has been the subject of debate for centuries. According to some theories, their leaders at least may have spoken a Turkic language, perhaps closest to the modern Chuvash language. According to the Encyclopedia of European Peoples, "the Huns, especially those who migrated to the west, may have been a combination of central Asian Turkic, Mongolic, and Ugric stocks".

Attila

841

Early life and background

Attila's father Mundzuk was the brother of kings Octar and Ruga, who reigned jointly over the Hunnic empire in the early fifth century. This form of diarchy was recurrent with the Huns, but historians are unsure whether it was institutionalized, merely customary, or an occasional occurrence. His family was from a noble lineage, but it is uncertain whether they constituted a royal dynasty. Attila's birthdate is debated; journalist Éric Deschodt and writer Herman Schreiber have proposed a date of 395. However, historian Iaroslav Lebedynsky and archaeologist Katalin Escher prefer an estimate between the 390s and the first decade of the fifth century. Several historians have proposed 406 as the date.

Attila

841

Early life and background

Attila grew up in a rapidly changing world. His people were nomads who had only recently arrived in Europe. They crossed the Volga river during the 370s and annexed the territory of the Alans, then attacked the Gothic kingdom between the Carpathian mountains and the Danube. They were a very mobile people, whose mounted archers had acquired a reputation for invincibility, and the Germanic tribes seemed unable to withstand them. Vast populations fleeing the Huns moved from Germania into the Roman Empire in the west and south, and along the banks of the Rhine and Danube. In 376, the Goths crossed the Danube, initially submitting to the Romans but soon rebelling against Emperor Valens, whom they killed in the Battle of Adrianople in 378. Large numbers of Vandals, Alans, Suebi, and Burgundians crossed the Rhine and invaded Roman Gaul on December 31, 406 to escape the Huns. The Roman Empire had been split in half since 395 and was ruled by two distinct governments, one based in Ravenna in the West, and the other in Constantinople in the East. The Roman Emperors, both East and West, were generally from the Theodosian family in Attila's lifetime (despite several power struggles).

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841

Early life and background

The Huns dominated a vast territory with nebulous borders determined by the will of a constellation of ethnically varied peoples. Some were assimilated to Hunnic nationality, whereas many retained their own identities and rulers but acknowledged the suzerainty of the king of the Huns. The Huns were also the indirect source of many of the Romans' problems, driving various Germanic tribes into Roman territory, yet relations between the two empires were cordial: the Romans used the Huns as mercenaries against the Germans and even in their civil wars. Thus, the usurper Joannes was able to recruit thousands of Huns for his army against Valentinian III in 424. It was Aëtius, later Patrician of the West, who managed this operation. They exchanged ambassadors and hostages, the alliance lasting from 401 to 450 and permitting the Romans numerous military victories. The Huns considered the Romans to be paying them tribute, whereas the Romans preferred to view this as payment for services rendered. The Huns had become a great power by the time that Attila came of age during the reign of his uncle Ruga, to the point that Nestorius, the Patriarch of Constantinople, deplored the situation with these words: "They have become both masters and slaves of the Romans".

Attila

841

Campaigns against the Eastern Roman Empire

The death of Rugila (also known as Rua or Ruga) in 434 left the sons of his brother Mundzuk, Attila and Bleda, in control of the united Hun tribes. At the time of the two brothers' accession, the Hun tribes were bargaining with Eastern Roman Emperor Theodosius II's envoys for the return of several renegades who had taken refuge within the Eastern Roman Empire, possibly Hunnic nobles who disagreed with the brothers' assumption of leadership.

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841

Campaigns against the Eastern Roman Empire

The following year, Attila and Bleda met with the imperial legation at Margus (Požarevac), all seated on horseback in the Hunnic manner, and negotiated an advantageous treaty. The Romans agreed to return the fugitives, to double their previous tribute of 350 Roman pounds (c. 115 kg) of gold, to open their markets to Hunnish traders, and to pay a ransom of eight solidi for each Roman taken prisoner by the Huns. The Huns, satisfied with the treaty, decamped from the Roman Empire and returned to their home in the Great Hungarian Plain, perhaps to consolidate and strengthen their empire. Theodosius used this opportunity to strengthen the walls of Constantinople, building the city's first sea wall, and to build up his border defenses along the Danube.

Attila

841

Campaigns against the Eastern Roman Empire

The Huns remained out of Roman sight for the next few years while they invaded the Sassanid Empire. They were defeated in Armenia by the Sassanids, abandoned their invasion, and turned their attentions back to Europe. In 440, they reappeared in force on the borders of the Roman Empire, attacking the merchants at the market on the north bank of the Danube that had been established by the treaty of 435.