134 Sentences With "ordinal numbers" | Random Sentence Generator

For ordinal numbers (numbers indicating position) greater than ten the cardinal is used.

The successor points and zero are the isolated points of the class of ordinal numbers, with respect to the order topology..

Cardinal and ordinal numbers must agree in gender (masculine or feminine; mixed groups are treated as masculine) with the noun they are describing. If there is no such noun (e.g. a telephone number or a house number in a street address), the feminine form is used. Ordinal numbers must also agree in number and definite status like other adjectives.

Roman numerals above string instrument notes. Stackexchange (2017). The position can be indicated by ordinal numbers (e.g., "3rd") or a roman numeral (e.g.

Ordinal numbers (first, second, third, etc.) are formed by preceding the number with ic or inic.Andrews (2001): p. 452; Lockhart (2001): p. 50.

324 & footnote (c): "This would appear more like a restitution of the old dignity than the creation of a new earldom"; Debrett's Peerage however gives the ordinal numbers as if a new earldom had been created. (Montague-Smith, P.W. (ed.), Debrett's Peerage, Baronetage, Knightage and Companionage, Kelly's Directories Ltd, Kingston- upon-Thames, 1968, p.353) and thus alternative ordinal numbers exist, given here.

For dealing with infinite sets, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers. The former gives the ordering of the set, while the latter gives its size. For finite sets, both ordinal and cardinal numbers are identified with the natural numbers. In the infinite case, many ordinal numbers correspond to the same cardinal number.

324 & footnote (c): "This would appear more like a restitution of the old dignity than the creation of a new earldom"; Debrett's Peerage however gives the ordinal numbers as if a new earldom had been created. (Montague-Smith, P.W. (ed.), Debrett's Peerage, Baronetage, Knightage and Companionage, Kelly's Directories Ltd, Kingston-upon-Thames, 1968, p.353) and thus alternative ordinal numbers exist, given here.

In mathematics, infinity plus one has meaning for the hyperreals, and also as the number ω+1 (omega plus one) in the ordinal numbers and surreal numbers.

However, S may be bounded as subset of Rn with the lexicographical order, but not with respect to the Euclidean distance. A class of ordinal numbers is said to be unbounded, or cofinal, when given any ordinal, there is always some element of the class greater than it. Thus in this case "unbounded" does not mean unbounded by itself but unbounded as a subclass of the class of all ordinal numbers.

Ordinal numbers written in digits take a full stop (e.g. 3. sor '3rd line'). The full stop is retained even before the hyphen that connects suffixes (e.g. a 10.

Cardinal and Ordinal Numbers is a book on transfinite numbers, by Polish mathematician Wacław Sierpiński. It was published in 1958 by Państwowe Wydawnictwo Naukowe, as volume 34 of the series Monografie Matematyczne of the . Sierpiński wrote on the same topic earlier, in his 1928 book Leçons sur les nombres tranfinis, but his 1958 book on the topic was completely rewritten and significantly longer. A second edition of Cardinal and Ordinal Numbers was published in 1965.

The integrity of the von Neumann universe depends fundamentally on the integrity of the ordinal numbers, which act as the rank parameter in the construction, and the integrity of transfinite induction, by which both the ordinal numbers and the von Neumann universe are constructed. The integrity of the ordinal number construction may be said to rest upon von Neumann's 1923 and 1928 papers., . See also the English-language presentation of von Neumann's "general recursion theorem" by .

Within set theory, many collections of sets turn out to be proper classes. Examples include the class of all sets, the class of all ordinal numbers, and the class of all cardinal numbers. One way to prove that a class is proper is to place it in bijection with the class of all ordinal numbers. This method is used, for example, in the proof that there is no free complete lattice on three or more generators.

Ordinal numbers (in reverse word order) are used for naming ruling members of a monarchy and the Popes. For example: Carol al II- lea, Papa Benedict al XVI-lea. See above for details.

We will prove this by reductio ad absurdum. # Let \Omega be a set that contains all ordinal numbers. # \Omega is transitive because for every element x of \Omega (which is an ordinal number and can be any ordinal number) and every element y of x (i.e. under the definition of Von Neumann ordinals, for every ordinal number y < x), we have that y is an element of \Omega because any ordinal number contains only ordinal numbers, by the definition of this ordinal construction.

Noun ellipsis (also N-ellipsis, N'-ellipsis, NP-ellipsis, NPE, ellipsis in the DP) occurs when the noun and potentially accompanying modifiers are omitted from a noun phrase.See Lobeck 2006 for an overview. Nominal ellipsis occurs with a limited set of determinatives in English (cardinal and ordinal numbers and possessive determiners), whereas it is much freer in other languages. The following examples illustrate nominal ellipsis with cardinal and ordinal numbers: ::Fred did three onerous tasks because Susan had done two onerous tasks.

The version of the paradox above is anachronistic, because it presupposes the definition of the ordinals due to John von Neumann, under which each ordinal is the set of all preceding ordinals, which was not known at the time the paradox was framed by Burali-Forti. Here is an account with fewer presuppositions: suppose that we associate with each well-ordering an object called its order type in an unspecified way (the order types are the ordinal numbers). The order types (ordinal numbers) themselves are well- ordered in a natural way, and this well-ordering must have an order type \Omega. It is easily shown in naïve set theory (and remains true in ZFC but not in New Foundations) that the order type of all ordinal numbers less than a fixed \alpha is \alpha itself.

Centuries are named using ordinal numbers in reverse order: "14th century" is secolul al paisprezecelea (normally written secolul al XIV-lea). Cardinal numbers are often used although considered incorrect: secolul paisprezece. See above for details. Royal titles.

3,14 '3.14'). Numbers are usually written in Arabic numerals. Roman numerals are only used in some special traditional cases, only to express ordinal numbers (e.g. to express the numbering of monarchs, popes, districts of a city, congresses, etc.).

The ordinal numbers are regular adjectives in Slovene. They have only definite forms, so the masculine nominative singular ends in -i. In writing, ordinals may be written in digit form followed by a period, as in German: 1., 2.

Cantor then defines the addition and multiplication of the cardinal and ordinal numbers. In 1885, Cantor extended his theory of order types so that the ordinal numbers simply became a special case of order types. In 1891, he published a paper containing his elegant "diagonal argument" for the existence of an uncountable set. He applied the same idea to prove Cantor's theorem: the cardinality of the power set of a set A is strictly larger than the cardinality of A. This established the richness of the hierarchy of infinite sets, and of the cardinal and ordinal arithmetic that Cantor had defined.

So the order type of all ordinal numbers less than \Omega is \Omega itself. But this means that \Omega, being the order type of a proper initial segment of the ordinals, is strictly less than the order type of all the ordinals, but the latter is \Omega itself by definition. This is a contradiction. If we use the von Neumann definition, under which each ordinal is identified as the set of all preceding ordinals, the paradox is unavoidable: the offending proposition that the order type of all ordinal numbers less than a fixed \alpha is \alpha itself must be true.

Transfinite numbers: Numbers that are greater than any natural number. Ordinal numbers: Finite and infinite numbers used to describe the order type of well-ordered sets. Cardinal numbers: Finite and infinite numbers used to describe the cardinalities of sets. Infinitesimals: Nilpotent numbers.

Schröder also made original contributions to algebra, set theory, lattice theory,"The Algebra of Logic Tradition". Stanford Encyclopedia of Philosophy. ordered sets and ordinal numbers. Along with Georg Cantor, he codiscovered the Cantor–Bernstein–Schröder theorem, although Schröder's proof (1898) is flawed.

Numbers in Hlai language, including cardinal numbers, ordinal numbers, and numbers of approximation, usually act as subjects, predicate, or objects in a sentence. When numbers are used with classifiers, together they become a phrase that can be an attribute to modify the noun phrase.

If there is no such noun (e.g. telephone numbers), the feminine form is used. For ordinal numbers greater than ten the cardinal is used and numbers above the value 20 have no gender. Jewish Town Hall building in Prague, with Hebrew numerals in counterclockwise order.

In the mathematical field of category theory, FinSet is the category whose objects are all finite sets and whose morphisms are all functions between them. FinOrd is the category whose objects are all finite ordinal numbers and whose morphisms are all functions between them.

So instead of saying, for example, "three days", Hopi would say the equivalent of "on the third day", using ordinal numbers. Whorf argues that the Hopi do not consider the process of time passing to produce another new day, but merely as bringing back the daylight aspect of the world.

The aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number. If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs. Cardinality is studied for its own sake as part of set theory.

Many of the older streets are named for ordinal numbers (First Street through to Twentyfifth Street). There is no obvious pattern to the layout of streets or the order in which they are named. Some streets on the eastern side align to the contours where the land is too steep to build.

New Toronto Plant. In 1890, new streets for the Town of New Toronto were laid out in several series, essentially without names by simply using ordinal numbers (First, Second, Third, etc.). When the streets were laid out along Lake Shore Road (now Lake Shore Blvd. West), they had a single new starting point.

For example, the set of 3-tuples of elements from a 2-element set has cardinality . In cardinal arithmetic, κ0 is always 1 (even if κ is an infinite cardinal or zero). Exponentiation of cardinal numbers is distinct from exponentiation of ordinal numbers, which is defined by a limit process involving transfinite induction.

The following is a timeline of the Tenrikyo religion, highlighting significant events since the birth of Tenrikyo's foundress Miki Nakayama. Specific dates are provided in parentheses; the lunar calendar is indicated with ordinal numbers (e.g. 18th day of 4th month) while the Gregorian calendar is indicated with name and number (e.g. August 15).

The Hebrew language has names for common numbers that range from zero to one million. Letters of the Hebrew alphabet are used to represent numbers in a few traditional contexts, for example in calendars. In other situations Arabic numerals are used. Cardinal and ordinal numbers must agree in gender with the noun they are describing.

Ordinal-linguistic personification (OLP, or personification for short) is a form of synesthesia in which ordered sequences, such as ordinal numbers, days, months and letters are associated with personalities and/or genders (). Although this form of synesthesia was documented as early as the 1890s (; ) researchers have, until recently, paid little attention to this form (see History of synesthesia research).

Every spring, Siyuan will select excellent freshmen from all departments. After the writing test and interview, the list of new Siyuaners will be open to the public. Students graduated in different years are named after ordinal numbers. The first Siyuaners are the 2001 ones, who are called Siyuan 1st, so the most recent ones are called Siyuan 7th.

Ordinal numbers have grammatically no differences with adjectives. While forming them, upper three orders of numerals are agglutinated to nearest dividing power of 1000, which results in constructing some of the longest natural Russian words, e.g. (153,000-th), while the next is (153,001-st). In the latter example, only the last word is declined with noun.

An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size". Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted.

After the incorporation, some of its streets were given new names. In 1931, north-south street names were standardized by continuing the ordinal numbers of New Toronto's streets, picking-up at Twenty-Third Street in the east through to Forty-Third Street in the west. For example; Lansdowne Avenue became Thirty- Third Street and Lake View Avenue became Thirty-Fifth Street.

Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, are a system of writing numbers using the letters of the Greek alphabet. In modern Greece, they are still used for ordinal numbers and in contexts similar to those in which Roman numerals are still used elsewhere in the West. For ordinary cardinal numbers, however, Greece uses Arabic numerals.

Tait (2005) gives a game-theoretic interpretation of Gentzen's method. Gentzen's consistency proof initiated the program of ordinal analysis in proof theory. In this program, formal theories of arithmetic or set theory are assigned ordinal numbers that measure the consistency strength of the theories. A theory will be unable to prove the consistency of another theory with a higher proof theoretic ordinal.

However, the use of the adjective alone is fairly common in the case of superlatives such as biggest, ordinal numbers such as first, second, etc., and other related words such as next and last. Many adjectives, though, have undergone conversion so that they can be used regularly as countable nouns; examples include Catholic, Protestant, red (with various meanings), green, etc.

Class I adjectives for which the last syllable in the masculine direct singular form is ور /‑wár/, ګر /‑gár/, جن /‑ján/, or م ن /‑mán/, as well as ordinal numbers ending in م /‑ám/, undergo a different vowel alternation: the vowel /á/ of the final syllable centralizes to /ə́/ in feminine non-direct singulars and in all plural forms, irrespective of gender.

Lavielle and Morton was the first architecture firm west of the Mississippi River above New Orleans. As street commissioner in 1823–26, Joseph Laveille devised the city's street name grid, with ordinal numbers for north-south streets and arboreal names for east-west streets.Laveille and Morton - stlcin.missouri.org - Retrieved January 21, 2008 Missouri became a state in 1821, and the St. Louis population tripled in 10 years.

The cardinal numbers in Frater: 1 - uni 2 - bi 3 - tri 4 - kuadri 5 - kuinti 6 - ses 7 - sep 8 - okta 9 - nona 10 - deka 11 - dekauni 12 - dekabi 13 - dekatri 20 - bideka 24 - bidekakuadri 30 - trideka 40 - kuadrideka 85 - oktadekakuinti 100 - senti 367 - trisenti-sesdeka-sep 600 - sessenti 1000 - mil 1000000 - milion Ordinal numbers are formed by placing the cardinal number after the noun.

Unlike choice voting where the numbers represent the order of a voter's ranking of candidates (i.e. they are ordinal numbers), in cumulative votes the numbers represent quantities (i.e. they are cardinal numbers). While giving voters more points may appear to give them a greater ability to graduate their support for individual candidates, it is not obvious that it changes the democratic structure of the method.

Skepticism about the axiom of choice was reinforced by recently discovered paradoxes in naive set theory. Cesare Burali-Forti (1897) was the first to state a paradox: the Burali-Forti paradox shows that the collection of all ordinal numbers cannot form a set. Very soon thereafter, Bertrand Russell discovered Russell's paradox in 1901, and Jules Richard (1905) discovered Richard's paradox. Zermelo (1908b) provided the first set of axioms for set theory.

A field army is composed of 100,000 to 300,000 troops. Specific field armies are usually named or numbered to distinguish them from "army" in the sense of an entire national land military force. In English, the typical orthographic style for writing out the names field armies is word numbers, such as "First Army"; whereas corps are usually distinguished by Roman numerals (e.g. I Corps) and subordinate formations with ordinal numbers (e.g.

In set theory, there are exponential operations for cardinal and ordinal numbers. If κ and λ are cardinal numbers, the expression κλ represents the cardinality of the set of functions from any set of cardinality λ to any set of cardinality κ.Nicolas Bourbaki, Elements of Mathematics, Theory of Sets, Springer-Verlag, 2004, III.§3.5. If κ and λ are finite, then this agrees with the ordinary arithmetic exponential operation.

In the old days, Saigon's roads were simply named by ordinal numbers. Starting from the Saigon River bank, Đồng Khởi was the Sixth Road. In 1865, the French Commander Admiral De La Grandiere renamed these roads and Sixth Road became Rue Catinat, a bustling place. Across the street from the future Continental site, the first foundations and floors for factories were built, the first one for Denis Frere.

Philologists have debated the origin and meaning of these names since classical antiquity. However, many of the meanings popularly assigned to various praenomina appear to have been no more than "folk etymology". The names derived from numbers are the most certain. The masculine names Quintus, Sextus, Septimus, Octavius and Decimus, and the feminine names Prima, Secunda, Tertia, Quarta, Quinta, Sexta, Septima, Octavia, Nona and Decima are all based on ordinal numbers.

German grammar rules do not allow for leading zeros in dates at all, and there should always be a space after a dot. However, leading zeros were allowed according to machine writing standards if they helped aligning dates. The use of a dot as a separator matches the convention of pronouncing the day and the month as an ordinal number, because ordinal numbers are written in German followed by a dot.

In ordinal ranking, all items receive distinct ordinal numbers, including items that compare equal. The assignment of distinct ordinal numbers to items that compare equal can be done at random, or arbitrarily, but it is generally preferable to use a system that is arbitrary but consistent, as this gives stable results if the ranking is done multiple times. An example of an arbitrary but consistent system would be to incorporate other attributes into the ranking order (such as alphabetical ordering of the competitor's name) to ensure that no two items exactly match. With this strategy, if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first") and D gets ranking number 4 ("fourth"), and either B gets ranking number 2 ("second") and C gets ranking number 3 ("third") or C gets ranking number 2 ("second") and B gets ranking number 3 ("third").

After five introductory chapters on naive set theory and set-theoretic notation, and a sixth chapter on the axiom of choice, the book has four chapters on cardinal numbers, their arithmetic, and series and products of cardinal numbers, comprising approximately 50 pages. Following this, four longer chapters (totalling roughly 180 pages) cover orderings of sets, order types, well-orders, ordinal numbers, ordinal arithmetic, and the Burali-Forti paradox according to which the collection of all ordinal numbers cannot be a set. Three final chapters concern aleph numbers and the continuum hypothesis, statements equivalent to the axiom of choice, and consequences of the axiom of choice. The second edition makes only minor changes to the first except for adding footnotes concerning two later developments in the area: the proof by Paul Cohen of the independence of the continuum hypothesis, and the construction by Robert M. Solovay of the Solovay model in which all sets of real numbers are Lebesgue measurable.

Since the adoption of Arabic numerals, numbers have become written in Arabic numerals more and more often. Counters and ordinal numbers are typically written in Arabic numbers, such as 3人 (three people), 7月 (July, "seventh-month"), 20歳 (age 20), etc., although 三人、七月、and 二十歳 are also acceptable to write (albeit less common). However, numbers that are part of lexemes are typically written in kanji.

The Lithuanian calendar is unusual among Western countries in that neither the names of the months nor the names of the weekdays are derived from Greek or Norse mythology. They were formalized after Lithuania regained independence in 1918, based on historic names, and celebrate natural phenomena; three months are named for birds, two for trees, and the remainder for seasonal activities and features. The days of the week are simply ordinal numbers.

Because the class of ordinal numbers is well-ordered, there is a smallest infinite limit ordinal; denoted by ω (omega). The ordinal ω is also the smallest infinite ordinal (disregarding limit), as it is the least upper bound of the natural numbers. Hence ω represents the order type of the natural numbers. The next limit ordinal above the first is ω + ω = ω·2, which generalizes to ω·n for any natural number n.

The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. It has also been shown (in von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorphic to the maximal class surreal field; in theories without the axiom of global choice, this need not be the case, and in such theories it is not necessarily true that the surreals are a universal ordered field.

Sierpiński authored 724 papers and 50 books, mostly in Polish. His book Cardinal and Ordinal Numbers was originally published in English in 1958. Two books, Introduction to General Topology (1934) and General Topology (1952) were translated into English by Canadian mathematician Cecilia Krieger. Another book, Pythagorean Triangles (1954), was translated into English by Indian mathematician Ambikeshwar Sharma, published in 1962, and republished by Dover Books in 2003; it also has a Russian translation.

Three well- orderings on the set of natural numbers with distinct order types (top to bottom): \omega, \omega+5, and \omega+\omega. Every well-ordered set is order- equivalent to exactly one ordinal number. The ordinal numbers are taken to be the canonical representatives of their classes, and so the order type of a well-ordered set is usually identified with the corresponding ordinal. For example, the order type of the natural numbers is .

Let S0 = S and let Sk+1 be the derived set of Sk. If there is a finite number n for which Sn is finite, then all the coefficients are zero. Later, Lebesgue proved that if there is a countably infinite ordinal α such that Sα is finite, then the coefficients of the series are all zero. Cantor's work on the uniqueness problem famously led him to invent transfinite ordinal numbers, which appeared as the subscripts α in Sα .

Due to the frequent contacts made between the Li (黎族) and the Han (汉族) over a relatively lengthy stretch of time, the Hlai language has been influenced by the Chinese language and its grammar. As previously mentioned, the Hlai counting system for dates, ordinal numbers, and measurements have been influenced by Chinese. In this chapter, the Chinese influence in Hlai's word order of attribute phrases, verb-object-complement phrases, and interrogative sentences is discussed.

As part of her thesis work, in 1952, Morel found two different countable ordinal numbers whose squares are equal. After Wacław Sierpiński simplified her construction, they published it jointly. In 1955, Morel published a converse to the Knaster–Tarski theorem, according to which every incomplete lattice has an increasing function with no fixed point. Her 1965 paper with Thomas Frayne and Dana Scott, "Reduced direct products", provides the main definitions of reduced products in model theory.

This is an eponymous album as he used one of his stage names, Aleph-1. The concept of the album and its name, Aleph-1, derive from the theories of German mathematician Georg Cantor, who was a teacher in Halle, Saxony-Anhalt, Germany, a city, to which Alva Noto is deeply connected with through his family. In mathematical terms, \aleph_1 is the cardinality of the set of all countable ordinal numbers or a number of elements in endless successions.

Rosser (1939) formally identified the three notions-as-definitions: Kleene proposes Church's Thesis: This left the overt expression of a "thesis" to Kleene. In his 1943 paper Recursive Predicates and Quantifiers Kleene proposed his "THESIS I": (22) references Church 1936; (23) references Turing 1936–7 Kleene goes on to note that: (24) references Post 1936 of Post and Church's Formal definitions in the theory of ordinal numbers, Fund. Math. vol 28 (1936) pp.11–21 (see ref.

In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set which represents the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. In addition to these usual ordinal operations, there are also the "natural" arithmetic of ordinals and the nimber operations.

In set theory, Scott's trick is a method for giving a definition of equivalence classes for equivalence relations on a proper class (Jech 2003:65) by referring to levels of the cumulative hierarchy. The method relies on the axiom of regularity but not on the axiom of choice. It can be used to define representatives for ordinal numbers in ZF, Zermelo–Fraenkel set theory without the axiom of choice (Forster 2003:182). The method was introduced by .

The paradoxes of naive set theory can be explained in terms of the inconsistent tacit assumption that "all classes are sets". With a rigorous foundation, these paradoxes instead suggest proofs that certain classes are proper (i.e., that they are not sets). For example, Russell's paradox suggests a proof that the class of all sets which do not contain themselves is proper, and the Burali-Forti paradox suggests that the class of all ordinal numbers is proper.

The idea that the collection of all ordinal numbers cannot logically exist seems paradoxical to many. This is related to Cesare Burali-Forti's "paradox" which states that there can be no greatest ordinal number. All of these problems can be traced back to the idea that, for every property that can be logically defined, there exists a set of all objects that have that property. However, as in Cantor's argument (above), this idea leads to difficulties.

It is generally also the tone most often repeated in the piece, and finally the range delimits the upper and lower tones for a given mode. The eight modes can be further divided into four categories based on their final (finalis). Medieval theorists called these pairs maneriae and labeled them according to the Greek ordinal numbers. Those modes that have d, e, f, and g as their final are put into the groups protus, deuterus, tritus, and tetrardus respectively.

1: un; 2: du; 3: tri; 4: kwer; 5: pin; 6: ses; 7: sep; 8: oc; 9: nev; 10: des; 100: sunte; 1000: tilie. 357: trisunte pindes-sep. Ordinal numbers are formed by adding -i or -j (after a vowel): duj: "second"; trij: "third", kweri: "fourth", pini: "fifth"; the exception is pri: "first". Fractions are formed by adding -t to numbers: u trit: "a third", u kwert: "a fourth, a quarter"; the exception is mij: "half".

On string instruments, a string change is a change from playing on one string to another. This may also involve a simultaneous change in fingering and/or position (shift), all of which must be done skillfully to avoid noticeable string noise. String may be indicated through Roman numerals (I-IV) or simply the string's base note's letter (e.g. - A, E, G, etc.), fingering may be indicated through numbers for the fingers (1-4), and position may be indicated through ordinal numbers (e.g. 2nd).

Since there are only countably many notations, all ordinals with notations are exhausted well below the first uncountable ordinal ω1; their supremum is called Church–Kleene ω1 or ω1CK (not to be confused with the first uncountable ordinal, ω1), described below. Ordinal numbers below ω1CK are the recursive ordinals (see below). Countable ordinals larger than this may still be defined, but do not have notations. Due to the focus on countable ordinals, ordinal arithmetic is used throughout, except where otherwise noted.

The reason for this is that the rankings are ordinal scale numbers, and multiplication is not defined for ordinal numbers. The ordinal rankings only say that one ranking is better or worse than another, but not by how much. For instance, a ranking of "2" may not be twice as severe as a ranking of "1", or an "8" may not be twice as severe as a "4", but multiplication treats them as though they are. See Level of measurement for further discussion.

Most typewriters for Spanish and other Romance languages had keys that could enter _o_ and _a_ directly, as a shorthand intended to be used primarily with ordinal numbers, such as 1. _o_ for first. In computing, early 8-bit character sets as code page 437 for the original IBM PC (circa 1981) also had these characters. In ISO-8859-1 Latin-1, and later in Unicode, they were assigned to and are known as U+00AA FEMININE ORDINAL INDICATOR (ª) and U+00BA MASCULINE ORDINAL INDICATOR (º).

For example, they can be constructed by taking powersets, or they can be separated as subsets of sets already "given". This, he says, eliminates contradictory ideas like "the set of all sets" or "the set of all ordinal numbers". He disposes of the Russell paradox by means of this Theorem: "Every set M possesses at least one subset M_0 that is not an element of M ". Let M_0 be the subset of M for which, by AXIOM III, is separated out by the notion "x otin x".

The mobilization model for the Wehrmacht's active and reserve forces in multiple waves was first issued in the annual mobilization plan of 8 December 1938. The system initially had four waves, the first of which would be the peacetime army and the other three raised in anticipation of the invasion of Poland. The first wave (the peacetime army) consisted of divisions with ordinal numbers of one to 50. The second wave, reservists who had completed their compulsory training, consisted of divisions numbered 51 to 100.

The Principia covered only set theory, cardinal numbers, ordinal numbers, and real numbers. Deeper theorems from real analysis were not included, but by the end of the third volume it was clear to experts that a large amount of known mathematics could in principle be developed in the adopted formalism. It was also clear how lengthy such a development would be. A fourth volume on the foundations of geometry had been planned, but the authors admitted to intellectual exhaustion upon completion of the third.

There are certain large cardinals that cannot exist in the constructible universe (L) of any model of set theory. Nevertheless, the constructible universe contains all the ordinal numbers that the original model of set theory contains. This "paradox" can be resolved by noting that the defining properties of some large cardinals are not absolute to submodels. One example of such a nonabsolute large cardinal axiom is for measurable cardinals; for an ordinal to be a measurable cardinal there must exist another set (the measure) satisfying certain properties.

David White acted as an agent to receive payments from Menard on behalf of the Republic of Texas. White claimed that Menard made the payments, but it is not clear about the form of the payments and how much, if any, was forwarded to the Republic of Texas. John D. Groesbeck completed his orthogonal plan for Galveston in 1838. He named the eastwest streets according to letters from the alphabet, and used ordinal numbers for northsouth streets, though many of these streets were renamed.

The three four-axle narrow gauge railcars of BDŽ class 05 01-03 which were procured in 1941 had proven very good in operation on the Rhodope Railway. In order to cope with the increase in traffic after the Second World War, the BDŽ procured four diesel railcars from Ganz Works Budapest in 1952 again. They were similar in construction and appearance to the vehicles of 1941, but more powerfully motorized with a power of . Originally, they were given the same series designation 05 with the ordinal numbers 04-07.

The sequence is infinite—and this statement requires some proof. The proof depends on the observation that the English names of all ordinal numbers, except those that end in 2, must contain at least one "t".. Aronson's sequence is closely related to autograms . There are many generalizations of Aronson's sequence and research into the topic is ongoing. write that Aronson's sequence is "a classic example of a self-referential sequence"; however, they criticize it for being ambiguously defined due to the variation in naming of numbers over one hundred in different dialects of English.

In fact, Cantor's method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of.The biographical material in this article is mostly drawn from Dauben 1979. Grattan-Guinness 1971, and Purkert and Ilgauds 1985 are useful additional sources. Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive – even shocking – that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri PoincaréDauben 2004, p. 1.

Until the 19th century, this commission made most of the economic decisions of Great Britain (England, before the Act of Union 1707). However, starting during the 19th century, these positions became sinecure positions, with the First Lord serving almost invariably as Prime Minister, the Second Lord invariably as Chancellor of the Exchequer, and the junior lords serving as whips in Parliament. As an office in commission, technically all Lords Commissioners of the Treasury are of equal rank, with their ordinal numbers connoting seniority rather than authority over their fellow Lords Commissioners.

The letter i in the word zeci (both as a separate word and in compounds), although thought by native speakers to indicate an independent sound, is only pronounced as a palatalization of the previous consonant. It does not form a syllable by itself: patruzeci "forty" is pronounced . The same applies to the last i in cinci: , including compounds: 15 is pronounced and 50 is . However, in the case of ordinal numbers in the masculine form, before -lea the nonsylabic i becomes a full syllabic i in words like douăzecilea "20th" and in cincilea "5th" .

It was used for the universe of sets in 1889 by Peano, the letter V signifying "Verum", which he used both as a logical symbol and to denote the class of all individuals.. See pages VIII and XI. Peano's notation V was adopted also by Whitehead and Russell for the class of all sets in 1910.. See page 229. The V notation (for the class of all sets) was not used by von Neumann in his 1920s papers about ordinal numbers and transfinite induction. Paul Cohen. See page 88.

Just as cardinal numbers form a (class) semiring, so do ordinal numbers form a near-ring, when the standard ordinal addition and multiplication are taken into account. However, the class of ordinals can be turned into a semiring by considering the so-called natural (or Hessenberg) operations instead. In category theory, a 2-rig is a category with functorial operations analogous to those of a rig. That the cardinal numbers form a rig can be categorified to say that the category of sets (or more generally, any topos) is a 2-rig.

Representation of the ordinal numbers up to ωω. Each turn of the spiral represents one power of ω. Limit ordinals are those that are non-zero and have no predecessor, such as ω or ω2 In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ordinal γ such that β < γ < λ.

In mathematics, the nimbers, also called Grundy numbers, are introduced in combinatorial game theory, where they are defined as the values of heaps in the game Nim. The nimbers are the ordinal numbers endowed with nimber addition and nimber multiplication, which are distinct from ordinal addition and ordinal multiplication. Because of the Sprague–Grundy theorem which states that every impartial game is equivalent to a Nim heap of a certain size, nimbers arise in a much larger class of impartial games. They may also occur in partisan games like Domineering.

Hugh de Courtenay, 1st/9th Earl of Devon (14 September 1276 – 23 December 1340). of Tiverton Castle, Okehampton Castle, Plympton Castle and Colcombe Castle, all in Devon, feudal baron of Okehampton and feudal baron of Plympton, was an English nobleman. In 1335, forty-one years after the death of his second-cousin once removed Isabel de Redvers, suo jure 8th Countess of Devon (died 1293) he was officially declared Earl of Devon, although whether as a new creation or in succession to her is unknown, thus alternative ordinal numbers exist for this Courtenay earldom.

Mechanisms of entrapment include pinchouts and local changes of permeability – forms of stratigraphic traps – and structural traps such as oil-bearing units blockaded by unrelated, impermeable units put there by motion along faults. Three separate producing horizons, or vertical zones, are present in the Puente Formation, and are given ordinal numbers: First, Second, and Third zones. In addition to these zones, small pockets of oil have been found throughout the upper part of the Puente.DOGGR, 258–259 The average depth of the three zones from top to bottom is 900, 1,100, and 1,500 feet.

Minimum excluded values of subclasses of the ordinal numbers are used in combinatorial game theory to assign nim-values to impartial games. According to the Sprague–Grundy theorem, the nim-value of a game position is the minimum excluded value of the class of values of the positions that can be reached in a single move from the given position. Minimum excluded values are also used in graph theory, in greedy coloring algorithms. These algorithms typically choose an ordering of the vertices of a graph and choose a numbering of the available vertex colors.

Cantor wanted the second paper to include a proof of the continuum hypothesis, but had to settle for expositing his theory of well-ordered sets and ordinal numbers. Cantor attempts to prove that if A and B are sets with A equivalent to a subset of B and B equivalent to a subset of A, then A and B are equivalent. Ernst Schröder had stated this theorem a bit earlier, but his proof, as well as Cantor's, was flawed. Felix Bernstein supplied a correct proof in his 1898 PhD thesis; hence the name Cantor–Bernstein–Schröder theorem.

Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable.

The vast majority of Paraguay's passports has the traces of handwriting of Konstanty Rokicki, but there are also several passports filled with a different character. The most probable version is that they are filled either by Juliusz Kühl or Stefan Ryniewicz, himself an experienced consul. Passports were issued for Jewish citizens of Poland, the Netherlands, Slovakia and Hungary as well as for Jews deprived of their Germany citizenship. The ordinal numbers of passports found in the Silberschein archives in Yad Vashem suggest that at least three series of these documents had been produced, tallying altogether to least 1056 pieces.

The notation is a finite string of symbols that intuitively stands for an ordinal number. By representing the ordinal in a finite way, Gentzen's proof does not presuppose strong axioms regarding ordinal numbers. He then proves by transfinite induction on these ordinals that no proof can conclude in a contradiction. The method used in this proof can also be used to prove a cut elimination result for Peano arithmetic in a stronger logic than first-order logic, but the consistency proof itself can be carried out in ordinary first-order logic using the axioms of primitive recursive arithmetic and a transfinite induction principle.

From 1985 to 1990 a new style of poetry arrived and was called "Transfinite," a word already used by the German mathematician, Georg CantorGeorg Cantor established a hierarchy among infinite sets, as well as transfinite ordinal numbers and calculations for working with them. (1845–1918). For the poet, "the Transfinite is the union of the finite and infinite in a transcendent synthesis : Its domain is Illumination.François Brousse, Conference, Paris, 18 February 1994." Poetry is a path, a type of ascension to the most ideal metamorphosis for night pilgrimsFrançois Brousse, Les Pèlerins de la nuit (The Night Pilgrims).

Rankings are ordinal numbers that reflect only the athletes' relative positions, not their playing skill as measured by a standard yardstick. UTR, in contrast, rates each athlete on a single, standard metric. Therefore, tennis players' UTRs are largely independent of each other, aside from the algorithm's weighting of the strength of opponents who compete directly with the rated player. Nearly all tennis ranking systems use a "points per round" (PPR) method that assigns points depending on what round a player reaches in a given tournament, along with the rated "strength" of that tournament in terms of the players it accepts into the draw.

In this context using "zeroth" as an ordinal is not strictly correct, but a widespread habit in this profession. Other programming languages, such as Fortran or COBOL, have array subscripts starting with one, because they were meant as high-level programming languages, and as such they had to have a correspondence to the usual ordinal numbers which predate the invention of the zero by a long time. Pascal allows the range of an array to be of any ordinal type (including enumerated types). APL allows setting the index origin to 0 or 1 during runtime programatically.

114 There is also a decimal counting system, which has become relatively widely used, though less so in giving the time, ages, and dates (it features no ordinal numbers). This system originated in Patagonian Welsh and was subsequently introduced to Wales in the 1940s. Whereas 39 in the vigesimal system is ("four on fifteen on twenty") or even ("two twenty minus one"), in the decimal system it is ("three tens nine"). Although there is only one word for "one" (), it triggers the soft mutation () of feminine nouns, where possible, other than those beginning with "ll" or "rh".

Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set. The position of each element within the ordered set is also given by an ordinal number. In the case of a finite set, the basic operation of counting, to find the ordinal number of a particular object, or to find the object with a particular ordinal number, corresponds to assigning ordinal numbers one by one to the objects. The size (number of elements, cardinal number) of a finite set is equal to the order type.

In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game of a certain type is determined; that is, one of the two players has a winning strategy. They motivated AD by its interesting consequences, and suggested that AD could be true in the least natural model L(R) of a set theory, which accepts only a weak form of the axiom of choice (AC) but contains all real and all ordinal numbers.

An initial natural number is given, and players alternate choosing positive divisors of the initial number, but may not choose 1 or a multiple of a previously chosen divisor. This game models n-dimensional Chomp, where the initial natural number has n prime factors and the dimensions of the Chomp board are given by the exponents of the primes in its prime factorization. Ordinal Chomp is played on an infinite board with some of its dimensions ordinal numbers: for example a 2 × (ω + 4) bar. A move is to pick any block and remove all blocks with both indices greater than or equal the corresponding indices of the chosen block.

There was widespread public debate leading up to the celebrations of the year 2000 as to whether the beginning of that year should be understood (and celebrated) as the beginning of "the" new millennium. Historically, there has been debate around the turn of previous decades, centuries and millennia. The issue arises from the difference between the convention of using ordinal numbers to count years and millennia (as in "the third millennium"), or cardinally using "the two thousands". The first convention is common in English-speaking countries, but the latter is favoured in, for example, Sweden (tvåtusentalet, which translates literally as the two thousands period).

For infinite groups, one can continue the lower central series to infinite ordinal numbers via transfinite recursion: for a limit ordinal λ, define Gλ = ∩ { Gα : α < λ}. If Gλ = 1 for some ordinal λ, then G is said to be a hypocentral group. For every ordinal λ, there is a group G such that Gλ = 1, but Gα ≠ 1 for all α < λ, . If ω is the first infinite ordinal, then Gω is the smallest normal subgroup of G such that the quotient is residually nilpotent, that is, such that every non-identity element has a non-identity homomorphic image in a nilpotent group .

A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers, and the class of all sets, are proper classes in many formal systems. In Quine's set-theoretical writing, the phrase "ultimate class" is often used instead of the phrase "proper class" emphasising that in the systems he considers, certain classes cannot be members, and are thus the final term in any membership chain to which they belong. Outside set theory, the word "class" is sometimes used synonymously with "set".

As of the beginning of the 17th century, the running of the Treasury was frequently entrusted to a commission, rather than to a single individual. Since 1714, it has permanently been in commission. The commissioners have always since that date been referred to as Lords Commissioners of the Treasury, and adopted ordinal numbers to describe their seniority. Eventually in the middle of the same century, the First Lord of the Treasury came to be seen as the natural head of the overall ministry running the country, and, as of the time of Robert Walpole (Whig), began to be known, unofficially, as the Prime Minister.

In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction. It is named after Cesare Burali-Forti, who, in 1897, published a paper proving a theorem which, unknown to him, contradicted a previously proved result by Cantor. Bertrand Russell subsequently noticed the contradiction, and when he published it in his 1903 book Principles of Mathematics, he stated that it had been suggested to him by Burali-Forti's paper, with the result that it came to be known by Burali-Forti's name.

Counting in the everyday sense typically starts from one, so it assigns to each object the size of the initial segment with that object as last element. Note that these numbers are one more than the formal ordinal numbers according to the isomorphic order, because these are equal to the number of earlier objects (which corresponds to counting from zero). Thus for finite n, the expression "n-th element" of a well-ordered set requires context to know whether this counts from zero or one. In a notation "β-th element" where β can also be an infinite ordinal, it will typically count from zero.

The characters Bo, Meng, Zhong, Shu and Ji are originally ordinals used in courtesy names to indicate a person's rank among his or her siblings of the same gender who survived to adulthood. The eldest brother's courtesy name would be prefixed with the word "Bo" (or "Meng" if he was born to a secondary wife), the second with "Zhong", the youngest with "Shu", and the rest with "Ji". For instance, Confucius' courtesy name was Zhongni. As the power of the Three Huan became hereditary, the descendants of Duke Zhuang's brothers used the ordinal numbers as family names to distinguish their branches of the House of Ji.

In the United States Army, a battalion is a unit composed of a headquarters and two to six batteries, companies, or troops. They are normally identified by ordinal numbers (1st Battalion, 2nd Squadron, etc.) and normally have subordinate units that are identified by single letters (Battery A, Company A, Troop A, etc.). Battalions are tactical and administrative organizations with a limited capability to plan and conduct independent operations and are normally organic components of brigades, groups, or regiments. A U.S. Army battalion includes the battalion commander (lieutenant colonel), executive officer (major), command sergeant major (CSM), headquarters staff, and usually three to five companies, with a total of 300 to 1,000 (but typically 500 to 600) soldiers.

The original definition of ordinal numbers, found for example in the Principia Mathematica, defines the order type of a well-ordering as the set of all well-orderings similar (order-isomorphic) to that well-ordering: in other words, an ordinal number is genuinely an equivalence class of well-ordered sets. This definition must be abandoned in ZF and related systems of axiomatic set theory because these equivalence classes are too large to form a set. However, this definition still can be used in type theory and in Quine's axiomatic set theory New Foundations and related systems (where it affords a rather surprising alternative solution to the Burali-Forti paradox of the largest ordinal).

In modern American baseball, some batting positions have nicknames: "leadoff" for first, "cleanup" for fourth, and "last" for ninth. Others are known by the ordinal numbers or the term #-hole (3rd place hitter would be 3-hole). In similar fashion, the third, fourth, and fifth batters are often collectively referred to as the "heart" or "meat" of the batting order, while the seventh, eighth, and ninth batters are called the "bottom of the lineup," a designation generally referring both to their hitting position and to their typical lack of offensive prowess. At the start of each inning, the batting order resumes where it left off in the previous inning, rather than resetting to start with the #1 hitter again.

The ARIA Awards are given in four fields: ARIA Awards (for general and genre categories), Fine Arts, Artisan and Public Vote. With the exception of the Public Vote field, all award winners and nominees are determined by either a "voting academy" or a "judging school"; the nominees for the public voted categories are determined by ARIA with the public choosing the winner. In the following tables, all the categories are listed in order of the year they were first given; any box in the "last awarded" column that says "N/A" is a current award. The years are linked to their corresponding ceremony and the ordinal numbers beside the year correspond to the order they were presented.

In linguistics, a numeral (or number word) in the broadest sense is a word or phrase that describes a numerical quantity. Some theories of grammar use the word "numeral" to refer to cardinal numbers that act as a determiner to specify the quantity of a noun, for example the "two" in "two hats". Some theories of grammar do not include determiners as a part of speech and consider "two" in this example to be an adjective. Some theories consider "numeral" to be a synonym for "number" and assign all numbers (including ordinal numbers like the compound word "seventy-fifth") to a part of speech called "numerals"Charles Follen: A Practical Grammar of the German Language.

Ordinal-linguistic personification (OLP, or personification for short) is a form of synesthesia in which ordered sequences, such as ordinal numbers, week-day names, months and alphabetical letters are associated with personalities or genders (). For example, the number 2 might be a young boy with a short temper, or the letter G might be a busy mother with a kind face. Although this form of synesthesia was documented as early as the 1890s (; ) researchers have, until recently, paid little attention to this form (see History of synesthesia research). This form of synesthesia was named as OLP in the contemporary literature by Julia Simner and colleagues although it is now also widely recognised by the term "sequence-personality" synesthesia.

The ordinal number given to the early Courtenay Earls of Devon depends on whether the earldom is deemed a new creation by the letters patent granted 22 February 1334/5 or whether it is deemed a restitution of the old dignity of the de Redvers family. Authorities differ in their opinions,Watson, in Cokayne, The Complete Peerage, new edition, IV, p.324 & footnote (c): "This would appear more like a restitution of the old dignity than the creation of a new earldom"; Debrett's Peerage however gives the ordinal numbers as if a new earldom had been created. (Montague-Smith, P.W. (ed.), Debrett's Peerage, Baronetage, Knightage and Companionage, Kelly's Directories Ltd, Kingston-upon-Thames, 1968, p.

A natural number (which, in this context, includes the number 0) can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. When restricted to finite sets, these two concepts coincide, and there is only one way to put a finite set into a linear sequence (up to isomorphism). When dealing with infinite sets, however, one has to distinguish between the notion of size, which leads to cardinal numbers, and the notion of position, which leads to the ordinal numbers described here. This is because while any set has only one size (its cardinality), there are many nonisomorphic well-orderings of any infinite set, as explained below.

Two well-orderings W_1 and W_2 are similar and write W_1 \sim W_2 just in case there is a bijection f from the field of W_1 to the field of W_2 such that x W_1 y \leftrightarrow f(x)W_2f(y) for all x and y. Similarity is shown to be an equivalence relation in much the same way that equinumerousness was shown to be an equivalence relation above. In New Foundations (NFU), the order type of a well-ordering W is the set of all well-orderings which are similar to W. The set of ordinal numbers is the set of all order types of well-orderings. This does not work in ZFC, because the equivalence classes are too large.

Shakespeare and Cervantes seemingly died on exactly the same date (23 April 1616), but Cervantes predeceased Shakespeare by ten days in real time (as Spain used the Gregorian calendar, but Britain used the Julian calendar). This coincidence encouraged UNESCO to make 23 April the World Book and Copyright Day. Astronomers avoid this ambiguity by the use of the Julian day number. For dates before the year 1, unlike the proleptic Gregorian calendar used in the international standard ISO 8601, the traditional proleptic Gregorian calendar (like the Julian calendar) does not have a year 0 and instead uses the ordinal numbers 1, 2, ... both for years AD and BC. Thus the traditional time line is 2 BC, 1 BC, AD 1, and AD 2.

As a result, subsequent formal treatments of calculus tended to drop the infinitesimal viewpoint in favor of limits, which can be performed using the standard reals. Infinitesimals regained popularity in the 20th century with Abraham Robinson's development of nonstandard analysis and the hyperreal numbers, which showed that a formal treatment of infinitesimal calculus was possible, after a long controversy on this topic by centuries of mathematics. Following this was the development of the surreal numbers, a closely related formalization of infinite and infinitesimal numbers that includes both the hyperreal numbers and ordinal numbers, and which is the largest ordered field. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though these entities were infinitely small.

Cello first position fingerings Fingered music for guitar: the numbers 1 to 4 indicate the stopping fingers, 0 an open note, circled numbers strings, and dashed numbers slipping On string instruments fingers are numbered from 1 to 4, beginning with the index finger, the thumb not being counted because it does not normally play on a string, and 0 indicating an open string. In those cases on string instruments where the thumb is used (such as high notes on a cello in thumb position), it is represented by a symbol the shape of an O with a vertical stem below(somewhat similar to Ǫ or ϙ, for instance). Guitar music indicates thumb, occasionally used to finger bass notes on the low E string, with a 'T'. Position may be indicated through ordinal numbers (e.g.

He could show that this proposition can neither be proved nor disproved within the formalism. This can mean only two things: either the reasoning by which a proof of consistency is given must contain some argument that has no formal counterpart within the system, i.e., we have not succeeded in completely formalizing the procedure of mathematical induction; or hope for a strictly "finitistic" proof of consistency must be given up altogether. When G. Gentzen finally succeeded in proving the consistency of arithmetic he trespassed those limits indeed by claiming as evident a type of reasoning that penetrates into Cantor's "second class of ordinal numbers." made the following comment in 1952 on the significance of Gentzen's result, particularly in the context of the formalist program which was initiated by Hilbert.

Hugh de Courtenay, 4th/12th Earl of Devon (1389 – 16 June 1422) was an English nobleman, son of the 3rd/11th Earl of Devon, and father of the 5th/13th Earl. The ordinal number given to the early Courtenay Earls of Devon depends on whether the earldom is deemed a new creation by the letters patent granted 22 February 1334/5 or whether it is deemed a restitution of the old dignity of the de Redvers family. Authorities differ in their opinions,Watson, in Cokayne, The Complete Peerage, new edition, IV, p.324 & footnote (c): "This would appear more like a restitution of the old dignity than the creation of a new earldom"; Debrett's Peerage however gives the ordinal numbers as if a new earldom had been created.

For ordinal numbers, when the numerals are preceded by the prefix tē (第), the colloquial set is used with the exception of numeral 1 and 2; when the numerals are preceded by the prefix thâu (頭), there is no exception to use the colloquial set when the number is smaller than 10, but once the number is greater than 10, the exception of numeral 1 and 2 appears again. Note that the system with prefix thâu is usually added by counter words, and it means "the first few"; for example, thâu-gō͘ pái means "the first five times". Thâu-chhit (number seven) sometimes means thâu-chhit kang (first seven days). It means the first seven days after a person died, which is a Hokkien cultural noun that should usually be avoided.

This means that, for every set S of cardinality \kappa, and every partition of the ordered pairs of elements of S into two subsets P_1 and P_1, there exists either a subset S_1\subset S of cardinality \kappa or a subset S_2\subset S of cardinality \alef_0, such that all pairs of elements of S_i belong to P_i. Here, P_1 can be interpreted as the edges of a graph having S as its vertex set, in which S_1 (if it exists) is a clique of cardinality \kappa, and S_2 (if it exists) is a countably infinite independent set. If S is taken to be the cardinal number \kappa itself, the theorem can be formulated in terms of ordinal numbers with the notation \kappa\rightarrow(\kappa,\omega)^2, meaning that S_2 (when it exists) has order type \omega.

Since the cardinal numbers are well-ordered by indexing with the ordinal numbers (see Cardinal number, formal definition), this also establishes that there is no greatest ordinal number; conversely, the latter statement implies Cantor's paradox. By applying this indexing to the Burali-Forti paradox we obtain another proof that the cardinal numbers are a proper class rather than a set, and (at least in ZFC or in von Neumann–Bernays–Gödel set theory) it follows from this that there is a bijection between the class of cardinals and the class of all sets. Since every set is a subset of this latter class, and every cardinality is the cardinality of a set (by definition!) this intuitively means that the "cardinality" of the collection of cardinals is greater than the cardinality of any set: it is more infinite than any true infinity. This is the paradoxical nature of Cantor's "paradox".

The project, which was launched over the course of March 2015, is > being developed purposely through a range of thirteenth sub-projects > represented graphically by a logo, a ring-shape dodecagon. The circle added > to the twelve-sided polygon symbolizes the thirteeth line. The graphic > choice, driven by the age of the band, is without a doubt well-thought since > a twelve-star polygon is culturally linked to the Earthly Branches, an > ancient means through which time is measured (duration, age). These sub- > projects, commonly titled ‘movement’ — the notion of time is a corollary of > the notion of movement — yet dissociated by ordinal numbers, are then > revealed in further details in dribs and drabs. Album ‘DOGMA’, being the > very first movement (located at the far north on the logo), is the core > piece of the puzzle which encompasses other audiovisual and text elements > created in association with 18 artists.

In Finnish orthography, when the numeral is followed by its head noun (which indicates the grammatical case of the ordinal), it is sufficient to write a period or full stop after the numeral: "In the competition, I finished in 2nd place". However, if the head noun is omitted, the ordinal indicator takes the form of a morphological suffix, which is attached to the numeral with a colon. In the nominative case, the suffix is for 1 and 2, and for larger numerals: "I came 2nd, and my brother came 3rd". This is derived from the endings of the spelled-out ordinal numbers: , , , , , , ... The system becomes rather complicated when the ordinal needs to be inflected, as the ordinal suffix is adjusted according to the case ending: (nominative case, which has no ending), (genitive case with ending ), (partitive case with ending ), (inessive case with ending ), (illative case with ending ), etc.

By contrast, a breadth-first search will never reach the grandchildren, as it seeks to exhaust the children first. A more sophisticated analysis of running time can be given via infinite ordinal numbers; for example, the breadth-first search of the depth 2 tree above will take ω·2 steps: ω for the first level, and then another ω for the second level. Thus, simple depth-first or breadth-first searches do not traverse every infinite tree, and are not efficient on very large trees. However, hybrid methods can traverse any (countably) infinite tree, essentially via a diagonal argument ("diagonal"—a combination of vertical and horizontal—corresponds to a combination of depth and breadth). Concretely, given the infinitely branching tree of infinite depth, label the root (), the children of the root (1), (2), …, the grandchildren (1, 1), (1, 2), …, (2, 1), (2, 2), …, and so on.

In 1335, forty-one years after the death of his second-cousin once removed Isabel de Redvers, suo jure 8th Countess of Devon (died 1293) (eldest daughter of Baldwin de Redvers, 6th Earl of Devon), letters patent were granted by King Edward III of England, dated 22 February 1335, declaring him Earl of Devon, and stating that he 'should assume such title and style as his ancestors, Earls of Devon, had wont to do so'. This thus made him 1st Earl of Devon, if the letters patent are deemed to have created a new peerage, otherwise 9th Earl of Devon, if it is deemed a restitution of the old dignity of the de Redvers family, and he is deemed to have succeeded the suo jure 8th Countess of Devon. Authorities differ in their opinions,Watson, in Cokayne, The Complete Peerage, new edition, IV, p.324 & footnote (c): "This would appear more like a restitution of the old dignity than the creation of a new earldom"; Debrett's Peerage however gives the ordinal numbers as if a new earldom had been created.

The Czech Republic hosts two different standardized forms of date and time writing. The Rules of Czech Orthography are mandatory for the educational system. These rules are based on tradition and are widely used by common people. The date is written in "day month year" order, each part separated by a space. Day and month are written as ordinal numbers and year as a cardinal number (1. 12. 2009). The month can be replaced by its full name in genitive case (1. prosince 2009). Writing the month in Roman digits (1. XII. 2009) is considered archaic. The time of day format is dot separated hours and minutes without a space (3.15). However to express time period the colon must be used (3:15). The second format is defined by the Czech State Norm (ČSN 01 6910) based on ISO standards. It accepts the ISO format (2009-12-01 and 03:15) and allows simplified traditional formatting and/or globalised formatting such as leading zeroes or omitted spaces (01.12.2009).

If α is a limit ordinal and X is a set, an α-indexed sequence of elements of X merely means a function from α to X. This concept, a transfinite sequence or ordinal-indexed sequence, is a generalization of the concept of a sequence. An ordinary sequence corresponds to the case α = ω. If X is a topological space, we say that an α-indexed sequence of elements of X converges to a limit x when it converges as a net, in other words, when given any neighborhood U of x there is an ordinal β<α such that xι is in U for all ι≥β. Ordinal-indexed sequences are more powerful than ordinary (ω-indexed) sequences to determine limits in topology: for example, ω1 (omega-one, the set of all countable ordinal numbers, and the smallest uncountable ordinal number), is a limit point of ω1+1 (because it is a limit ordinal), and, indeed, it is the limit of the ω1-indexed sequence which maps any ordinal less than ω1 to itself: however, it is not the limit of any ordinary (ω-indexed) sequence in ω1, since any such limit is less than or equal to the union of its elements, which is a countable union of countable sets, hence itself countable.