112 Sentences With "ordinal number" | Random Sentence Generator

RANK Rank is the ordinal number (first second third, etc.) that describes the position in a data set.

In set theory, the successor of an ordinal number α is the smallest ordinal number greater than α. An ordinal number that is a successor is called a successor ordinal.

This is a list of armies arranged by ordinal number.

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.

This is a list of military corps arranged by ordinal number.

In Chinese and Japanese, an ordinal number is prefixed by / ; for example, "first", "second".

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.

The article provides links to lists of military divisions arranged by ordinal number, name, country or conflict.

The article provides links to lists of military corps arranged by ordinal number, name, country or conflict.

In 1943, another corps-level unit carrying the ordinal number 21 was created, the XXI Mountain Corps.

These sets are then taken to "be" cardinal numbers, by definition. In Zermelo-Fraenkel set theory with the axiom of choice, one way of assigning representatives to cardinal numbers is to associate each cardinal number with the least ordinal number of the same cardinality. These special ordinals are the ℵ numbers. But if the axiom of choice is not assumed, for some cardinal numbers it may not be possible to find such an ordinal number, and thus the cardinal numbers of those sets have no ordinal number as representatives.

The rank among well-ordered sets is expressed by an ordinal number; for the natural numbers, this is denoted as (omega).

Divisions are commonly designated by combining an ordinal number and a type name (e.g.: "13th Infantry Division"). Nicknames are often assigned or adopted, although these often are not considered an official part of the unit's nomenclature, with divisions of the Italian Army being one of the exceptions. In some cases, divisional titles lack an ordinal number, often in the case of unique units or units serving as elite or special troops.

In 1936, Gentzen published a proof that Peano Arithmetic is consistent. Gentzen's result shows that a consistency proof can be obtained in a system that is much weaker than set theory. Gentzen's proof proceeds by assigning to each proof in Peano arithmetic an ordinal number, based on the structure of the proof, with each of these ordinals less than ε0.Actually, the proof assigns a "notation" for an ordinal number to each proof.

A hundredth is written as a decimal fraction as 0.01, and as a vulgar fraction as 1/100. “Hundredth” is also the ordinal number that follows “ninety-ninth” and precedes “hundred and first.” It is written as 100th.

Cars of the 20x class would be smaller than cars of the 30x or 40x class, and cars of the same class would be usually replaced by the next ordinal number (e.g. the 201 was replaced with the Peugeot 202).

The District Courts of Appeal originally consisted of three appellate districts, headquartered in San Francisco, Los Angeles, and Sacramento, with three justices each. These first nine justices were appointed by the Governor. Each district was assigned an ordinal number (i.e., first, second, and third).

Joannicius III (, ), (c. 1700 – 1793) was Archbishop of Peć and Serbian Patriarch from 1739 to 1746 and Archbishop of Constantinople and Ecumenical Patriarch from 1761 to 1763. The ordinal number of his title is III both for his office as Serbian Patriarch and of Constantinople.

In order-theoretic mathematics, the deviation of a poset is an ordinal number measuring the complexity of a partially ordered set. The deviation of a poset is used to define the Krull dimension of a module over a ring as the deviation of its poset of submodules.

Soviet and Russian staffed drifting ice stations are research stations built on the ice of the high latitudes of the Arctic Ocean. They are important contributors to exploration of the Arctic. The stations are named North Pole (NP; , ), followed by an ordinal number: North Pole-1, etc.

Examples: (first grade (in elementary school)), (third edition), but . Furthermore, suffixes can be left out if the number obviously is an ordinal number, example: (3rd ed). Using a full stop as an ordinal indicator is considered archaic, but still occurs in military contexts. Example: (5th company).

One motivation for the ZFC axioms is the cumulative hierarchy of sets introduced by John von Neumann., section 2. In this viewpoint, the universe of set theory is built up in stages, with one stage for each ordinal number. At stage 0 there are no sets yet.

This typical setup can be modified in various ways. For example, instead of being a subset of X, each move might consist of a pair (I, p) where I \subset X and p \in x. Alternatively, the sequence of moves might have length some ordinal number other than ω1.

101 (one hundred [and] one) is the natural number following 100 and preceding 102. It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one". As an ordinal number, 101st (one hundred [and] first), rather than 101th, is the correct form.

There have been several World Camps held by the Girl Guides and Girl Scouts, first held in 1924. Organized by the World Association of Girl Guides and Girl Scouts, unlike World Scout Jamborees, World Camps are not named with an ordinal number, nor is there an attempt to hold them at regular intervals.

Every ordinal number is either zero, or a successor ordinal, or a limit ordinal. For example, ω, the smallest ordinal greater than every natural number is a limit ordinal because for any smaller ordinal (i.e., for any natural number) n we can find another natural number larger than it (e.g. n+1), but still less than ω.

In the beginning of 1941, XXI Army Corps, already effectively an army-level unit since its designation as Gruppe XXI, was fully replaced and had its organizational structure transferred to the newly formed Armee Norwegen. With the ordinal number 21 freed up for German army corps, a new corps with that number, XXI Mountain Corps, was created in 1943.

Rota of Pope Alexander III, AD 1175 The rota is one of the symbols used by the Pope to authenticate documents such as papal bulls. It is a cross inscribed in two concentric circles. Pope Leo IX was the first pope to use it. The four inner quadrants contain: "Petrus", "Paulus", the Pope's name, and the Pope's ordinal number.

Philadelphia's 10th Street written in English and Chinese A numbered street is a street whose name is an ordinal number, as in Second Street or Tenth Avenue. Such forms are among the most common street names in North America, but also exist in other parts of the world, especially in the Middle East. Numbered streets were first used in PhiladelphiaRybczynski, Witold. "City Life".

Every sentence must contain formal words to designate what Husserl calls "formal categories". There are two kinds of categories: meaning categories and formal-ontological categories. Meaning categories relate judgments; they include forms of conjunction, disjunction, forms of plural, among others. Formal-ontological categories relate objects and include notions such as set, cardinal number, ordinal number, part and whole, relation, and so on.

This is how this king is still referred to in Thai history books. His descendant Vajiravudh (Rama VI) who had studied in England, realised that most Siamese kings' names were difficult to reproduce and remember for Westerners. He therefore disposed to use for all kings of the Chakri dynasty the name Rama together with the respective ordinal number. So this king is Rama I in Western literature.

This process can be extended for all natural numbers n, and these are called n-categories. There is even a notion of ω-category corresponding to the ordinal number ω. Higher-dimensional categories are part of the broader mathematical field of higher-dimensional algebra, a concept introduced by Ronald Brown. For a conversational introduction to these ideas, see John Baez, 'A Tale of n-categories' (1996).

Two-Micron Sky Survey, or IRC, or Caltech infrared catalog is the astronomical catalogue of the infrared sources published in the 1969 by Neugebauer and Leighton. Catalogue index consists of two numbers - declination rounded to multiplier of 10 degrees, with sign, and star ordinal number within declination band. Catalog contains about 5000 objects between declinations +15 and -15 degrees. Most of the sources are M-type stars.

Nouns are split into independent nouns and verbal nouns. Na is the only independent noun that is used in the Lau language. It is only added to nouns when one is expressing relationships, or it is added to cardinal numbers to form an ordinal number. Pronouns are words that replace a noun in a sentence and can function by themselves as a noun phrase.

Cervini received all votes except of his own, which he gave to Gian Pietro Carafa. He retained his baptismal name, adding to it only an ordinal number (Marcellus II). On that same day, he was consecrated bishop of Rome by Cardinal Gian Pietro Carafa, bishop of Ostia e Velletri and Dean of the College of Cardinals, and crowned by Cardinal Francesco Pisani, Protodeacon of S. Marco.

Arabic numerals are kept as Arabic numerals: 635 fēnjī (, extension 635) ##According to 6.1.5.4, the dì () used in ordinal numerals is followed by a hyphen: dì-yī (, first), dì-356 (, 356th). The hyphen should not be used if the word in which dì () and the numeral appear does not refer to an ordinal number in the context. For example: Dìwǔ (, a Chinese compound surname).

After his death, it would be 423 years before another pope would choose a name with an ordinal number less than IV (John Paul I). Cervini was the maternal uncle of Robert Bellarmine. Cervini's father and Pope Clement VII were personal friends. Cervini served in the household of Cardinal Alessandro Farnese. When Farnese became Pope, Cervini served as his secretary and was employed on some diplomatic missions.

In the United States, dates are traditionally written in the "month-day-year" order, with neither increasing nor decreasing order of significance. This order is used in both the traditional all-numeric date (e.g., "1/21/16" or "01/21/2016") and the expanded form (e.g., "January 21, 2016"—usually spoken with the year as a cardinal number and the day as an ordinal number, e.g.

Standard map symbol for a numbered Army, the 'X's are not substituting the army's number A particular army can be named or numbered to distinguish it from military land forces in general. For example, the First United States Army and the Army of Northern Virginia. In the British Army it is normal to spell out the ordinal number of an army (e.g. First Army), whereas lower formations use figures (e.g.

Each stage is assigned an ordinal number. The lowest stage, stage 0, consists of all entities having no members. We assume that the only entity at stage 0 is the empty set, although this stage would include any urelements we would choose to admit. Stage n, n>0, consists of all possible sets formed from elements to be found in any stage whose number is less than n.

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.

Any ordinal number can be made into a topological space by endowing it with the order topology; this topology is discrete if and only if the ordinal is a countable cardinal, i.e. at most ω. A subset of ω + 1 is open in the order topology if and only if either it is cofinite or it does not contain ω as an element. See the Topology and ordinals section of the "Order topology" article.

The general rule is that (for 1 and 2) or (for all other numbers, except , et cetera, but including and ) is appended to the numeral. The reason is that and respectively end the ordinal number words. The ordinals for 1 and 2 may however be given an form ( and instead of and ) when used about a male person (masculine natural gender), and if so they are written and . When indicating dates, suffixes are never used.

Next to its upper course, on the two sides there were empty spaces where the remains of a market have been found. Being a shrine's approach, the avenue passes under three torii, or Shinto gates, called respectively Ichi no Torii (first gate), Ni no Torii (second gate) and San no Torii (third gate). The ordinal number decreases with the distance from the shrine, so the closest to Tsurugaoka Hachiman-gū is actually San no Torii.

In formal expressions, the ordinal number used before the word order refers to the highest order of derivative in the series expansion used in the approximation. The expressions: a zeroth-order approximation, a first-order approximation, a second-order approximation, and so forth are used as fixed phrases. The expression a zero order approximation is also common. Cardinal numerals are occasionally used in expressions like an order zero approximation, an order one approximation, etc.

Any ordinal number can be turned into a topological space by using the order topology. When viewed as a topological space, ω1 is often written as [0,ω1), to emphasize that it is the space consisting of all ordinals smaller than ω1. If the axiom of countable choice holds, every increasing ω-sequence of elements of [0,ω1) converges to a limit in [0,ω1). The reason is that the union (i.e.

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 .

In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by V, is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (ZFC), is often used to provide an interpretation or motivation of the axioms of ZFC. The rank of a well-founded set is defined inductively as the smallest ordinal number greater than the ranks of all members of the set.; ; .

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 .

For abstract large numbers the numeral suffix -zig (as in zwanzig = 20, vierzig = 40, sechzig = 60) is used like 'umpteen': Das habe ich schon zigmal gesagt! ('I already said so umpteen times'). An unknown ordinal number is was-weiß-ich-wievielte/r/s ('what do I know how many-th') or drölf (fictional integer whose name is a portmanteau of the words zwölf, 12, and dreizehn, 13). Exponents of 10 are also used as in English.

In mathematical logic, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra are called Borel sets. Each Borel set is assigned a unique countable ordinal number called the rank of the Borel set. The Borel hierarchy is of particular interest in descriptive set theory. One common use of the Borel hierarchy is to prove facts about the Borel sets using transfinite induction on rank.

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.

Hyperarithmetical theory studies those sets that can be computed from a computable ordinal number of iterates of the Turing jump of the empty set. This is equivalent to sets defined by both a universal and existential formula in the language of second order arithmetic and to some models of Hypercomputation. Even more general recursion theories have been studied, such as E-recursion theory in which any set can be used as an argument to an E-recursive function.

The 271st Infantry Division () was an infantry division of the German Heer during World War II. In total, three infantry formations used the ordinal number 271 within the Wehrmacht. The first 271st Infantry Division's deployment was aborted in May 1940, whereas the second iteration of the division saw its deployment completed in November 1943 and was destroyed in August 1944. Subsequently, a division designated 271st Volksgrenadier Division () was deployed in August 1944 and remained in combat until 1945.

383 – 390, 1967 Existence of a partition of the ordinal number \omega_2 into two colors with no monochromatic uncountable sequentially closed subset is independent of ZFC, ZFC + CH, and ZFC + ¬CH, assuming consistency of a Mahlo cardinal.Shelah, S., Proper and Improper Forcing, Springer 1992Schlindwein, Chaz, Shelah's work on non-semiproper iterations I, Archive for Mathematical Logic (47) 2008 pp. 579 – 606Schlindwein, Chaz, Shelah's work on non-semiproper iterations II, Journal of Symbolic Logic (66) 2001, pp.

Magnus Haakonsson (, ; 1 (or 3) May 1238 – 9 May 1280) was King of Norway (as Magnus VI) from 1263 to 1280 (junior king from 1257).Magnus 6 Håkonsson Lagabøte – utdypning (Store norske leksikon) One of his greatest achievements was the modernisation and nationalisation of the Norwegian law-code, after which he is known as Magnus the Law-mender (, ). He was the first Norwegian monarch known to have used an ordinal number, although originally counting himself as "IV".

During his patriarchate he spared no effort to improve the relations both with the Holy See and within the Chaldean Church, after the eventful reign of his predecessor Joseph Audo. He died in Mosul at the age of 54 on June 27, 1894. The ordinal number of his title is sometime XIV, sometime XIII, while among scholars Eliya XII is often preferred. This is due to the uncertain list of the Patriarchal line of Alqosh in the 16th and 17th century.

In set theory, a branch of mathematics, an additively indecomposable ordinal α is any ordinal number that is not 0 such that for any \beta,\gamma<\alpha, we have \beta+\gamma<\alpha. Additively indecomposable ordinals are also called gamma numbers. The additively indecomposable ordinals are precisely those ordinals of the form \omega^\beta for some ordinal \beta. From the continuity of addition in its right argument, we get that if \beta < \alpha and α is additively indecomposable, then \beta + \alpha = \alpha.

In general, the counter words mentioned above are cardinal numbers, in that they indicate quantity. To transform a counter word into an ordinal number that denotes position in a sequence, 目 me is added to the end of the counter. Thus "one time" would be translated as 一回 ikkai, whereas "the first time" would be translated as 一回目 ikkaime. This rule is inconsistent, however, as counters without the me suffix are often used interchangeably with cardinal and ordinal meanings.

The more recent shorter convention is that an act amending "Foo Act yyy1" will have short title "Foo (Amendment) Act yyy2". If a law is passed with the same title as another law passed in the same year, an ordinal number will be added to distinguish it from the others; this is particularly common for Finance Acts (Finance (No. 3) Act 2010) and commencement orders that bring parts of an Act into force (Environment Act 1995 (Commencement No.13) (Scotland) Order 1998).

The length of a branch is the ordinal that is order isomorphic to the branch. For each ordinal α, the α-th level of T is the set of all elements of T of height α. A tree is a κ-tree, for an ordinal number κ, if and only if it has height κ and every level has size less than the cardinality of κ. The width of a tree is the supremum of the cardinalities of its levels.

The twenty arrondissements are arranged in the form of a clockwise spiral (often likened to a snail shell), starting from the middle of the city, with the first on the Right Bank (north bank) of the Seine. In French, notably on street signs, the number is often given in Roman numerals. For example, the Eiffel Tower belongs to the VIIe arrondissement while Gare de l'Est is in the Xe arrondissement. In daily speech, people use only the ordinal number corresponding to the arrondissement, e.g.

So apartment 8º-D (not 8D) means the 8th floor (hence the character "º" meaning ordinal number), apartment D (counting in clockwise direction, for those who are in the floor entrance). But a very common form for buildings with three apartments per floor is, Esq.-Frt./Fte. (Frente, en: Front - for the apartment located between left and right)-Dto. These universal rules simplify finding an apartment in a building, particularly for blind people, who do not need to ask where a given apartment is.

It also has first tone when used as an ordinal number (or part of one), and when it is immediately followed by any digit (including another ; hence both syllables of the word yīyī and its compounds have first tone). # When is used between two reduplicated words, it may become neutral in tone (e.g. kànyikàn ("to take a look of")). The numbers qī ("seven") and bā ("eight") sometimes display similar tonal behavior as yī, but for most modern speakers they are always pronounced with first tone.

In fact the contested distinction between the seventh and eighth tones surrounds the very issue of tone sandhi (between glottal stop (-m) and low rising (-d) tones). High and high-falling tones (marked by -b and -j in the RPA orthography, respectively) trigger sandhi in subsequent words bearing particular tones. A frequent example can be found in the combination for numbering objects (ordinal number + classifier + noun): ib (one) + tus (classifier) + dev (dog) > ib tug dev (note tone change on the classifier from -s to -g).

One millionth is equal to 0.000 001, or 1 x 10−6 in scientific notation. It is the reciprocal of a million, and can be also written as 1/1 000 000.. Units using this fraction can be indicated using the prefix "micro-" from Greek, meaning "small".. Numbers of this quantity are expressed in terms of μ (the Greek letter mu).. "Millionth" can also mean the ordinal number that comes after the nine hundred, ninety-nine thousand, nine hundred, ninety-ninth and before the million and first..

Except for some very fluent speakers (like news anchors), even in English-language media, dates are also often read with a cardinal instead of an ordinal number (e.g. "January one" rather than "January first" or "January the first") even if the written form is the same. This is mostly because educated Filipinos were taught to count English numbers cardinally, thus it carried over to their style of reading dates. In reading the day-month-year date notation used by some areas in the government (e.g.

Provided there exists an order isomorphism between two well-ordered sets, the order isomorphism is unique: this makes it quite justifiable to consider the two sets as essentially identical, and to seek a "canonical" representative of the isomorphism type (class). This is exactly what the ordinals provide, and it also provides a canonical labeling of the elements of any well-ordered set. Every well-ordered set (S,<) is order-isomorphic to the set of ordinals less than one specific ordinal number under their natural ordering. This canonical set is the order type of (S,<).

This can be circumvented by the use of the Axiom of Choice to select a representative from each equivalence class to replace [x]_R, which will be at the same type as x, or by choosing a canonical representative if there is a way to do this without invoking Choice (the use of representatives is hardly unknown in ZFC, either). In NFU, the use of equivalence class constructions to abstract properties of general sets is more common, as for example in the definitions of cardinal and ordinal number below.

A programming bug confusing these two year numbers is probably the cause of some Android users of Twitter being unable to log in around midnight of 29 December 2014 UTC. The ISO week calendar relies on the Gregorian calendar, which it augments, to define the new year day (Monday of week 01). As a result, extra weeks are spread across the 400-year cycle in a complex, seemingly random pattern. There is no simple algorithm to determine whether a year has 53 weeks from its ordinal number alone.

The term Pentecost comes from the Greek () meaning "fiftieth". It refers to the festival celebrated on the fiftieth day after Passover, also known as the "Feast of Weeks" and the "Feast of 50 days" in rabbinic tradition. The Septuagint uses the term to refer to the "Feast of Pentecost" only twice, in the deuterocanonical Book of Tobit and 2 Maccabees. The Septuagint writers also used the word in two other senses: to signify the year of Jubilee (), an event which occurs every 50th year, and in several passages of chronology as an ordinal number.

The concepts of divine kingship and royal ceremonies are clear examples of the influence of Brahmanism. The Coronation of the Thai monarch are practiced more or less in its original form even up to the present reign. The Thai idea that the king is a reincarnation of the Hindu deity Vishnu was adopted from Indian tradition. (Though this belief no longer exists today, the tradition to call each Thai king of the present Chakri dynasty Rama (Rama is an incarnation of Vishnu) with an ordinal number, such as Rama I, Rama II etc.

Musée de l'Aventure Peugeot. The exhibit label (2012) states: « Jusqu'à présente le modèles Peugeot se suivaient dans une numérotation plus ou moins logique qui d’ailleurs n’était que peu utilise par la publicité. Par exemple le 190 S était présenté dans les brochures sous le nom « Le 5 CV Peugeot ». Avec la 201 une ère novelle commence.» Cars after the Type 190 use names three digits long with a zero in the middle, beginning with the 201, abandoning Peugeot's original procedure of naming each new model with a successive ordinal number.

A set A\, is called admissible if it is transitive and \langle A,\in \rangle is a model of Kripke–Platek set theory. An ordinal number α is called an admissible ordinal if Lα is an admissible set. The ordinal α is an admissible ordinal if and only if α is a limit ordinal and there does not exist a γ < α for which there is a Σ1(Lα) mapping from γ onto α. If M is a standard model of KP, then the set of ordinals in M is an admissible ordinal.

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.

A construction of the surreal numbers as a maximal binary pseudo-tree with simplicity (ancestor) and ordering relations is due to Philip Ehrlich, The difference from the usual definition of a tree is that the set of ancestors of a vertex is well-ordered, but may not have a maximal element (immediate predecessor); in other words the order type of that set is a general ordinal number, not just a natural number. This construction fulfills Alling's axioms as well and can easily be mapped to the sign-sequence representation.

A new bill to amend the constitution is usually named with the ordinal number next after that of the last amendment passed. Multiple pending bills will often use the same number, and be distinguished by year of introduction and/or a parenthetical number or description. However, if the government introduces multiple bills, these are numbered consecutively. There are several gaps in the numbering of passed amendments, corresponding to government bills which did not pass: ;Twelfth: Amendments 12, 13, and 14, all relating to abortion, were put to referendums on the same day.

For the precise definition, suppose that is a set of nodes. Using the reflexivity of partial orders, we can identify any tree on a subset of with its partial order - a subset of . The set of all relations that form a well-founded tree on a subset of is defined in stages , so that }. For each ordinal number , let belong to the -th stage if and only if is equal to : where is a subset of such that elements of are pairwise disjoint, and is a node that does not belong to .

First, if we take the powerset of any infinite set x, then that powerset will contain elements which are subsets of x of every finite cardinality (among other subsets of x). Proving the existence of those finite subsets may require either the axiom of separation or the axioms of pairing and union. Then we can apply the axiom of replacement to replace each element of that powerset of x by the initial ordinal number of the same cardinality (or zero, if there is no such ordinal). The result will be an infinite set of ordinals.

Some place names have become synonymous with battles, such as the Passchendaele, Pearl Harbor, the Alamo, Thermopylae and Waterloo. Military operations, many of which result in battle, are given codenames, which are not necessarily meaningful or indicative of the type or the location of the battle. Operation Market Garden and Operation Rolling Thunder are examples of battles known by their military codenames. When a battleground is the site of more than one battle in the same conflict, the instances are distinguished by ordinal number, such as the First and Second Battles of Bull Run.

One of the bank's shareholders became King Peter I Karađorđević. At the idea of King Peter I, in 1905, through the Board of Directors of Prometna Banka, "'Serbia', the first Serbian insurance company" was opened, ie the first insurance company in Serbia (except for the insurance department of the Belgrade Cooperative, which already existed). Shares with the ordinal number from 1 to 300 were also bought by King Peter I, which helped the idea of realizing such a society. From the very beginning, Savčić was in the management of that company and actively managed its affairs.

Poetto is popularly divided into "fermate" (stops), which means the various stretches of beach are recognized by the ordinal number of bus stops or urban lines linking the city centre district. The most popular is the 1st stop, adjacent to the port of Marina Piccola, just below the Sella del Diavolo. Very famous and popular are the 2nd (D'Aquila), 3rd (Lido) 4th and the 6th, home of an old Hospital and of major events (Championships of Beach Volleyball, Beach soccer, Beach Football and concerts). Around the 10th stop the coastline belonging to the Comune of Quartu Sant'Elena begins.

Dates. Calendar dates in Romanian are expressed using cardinal numbers, unlike English. For example, "the 21st of April" is 21 aprilie (read douăzeci și unu aprilie). For the first day of a month the ordinal number întâi is often used: 1 Decembrie (read Întâi Decembrie; upper case is used for names of national or international holidays). Normally the masculine form of the number is used everywhere, but when the units digit is 2, the feminine is also frequent: 2 ianuarie can be read both doi ianuarie and două ianuarie; the same applies for days 12 and 22. Centuries.

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 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.

The distinction between strict and non-strict well orders is often ignored since they are easily interconvertible. Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set. The well-ordering theorem, which is equivalent to the axiom of choice, states that every set can be well ordered. If a set is well ordered (or even if it merely admits a well-founded relation), the proof technique of transfinite induction can be used to prove that a given statement is true for all elements of the set.

In mathematics, the height of an element g of an abelian group A is an invariant that captures its divisibility properties: it is the largest natural number N such that the equation Nx = g has a solution x ∈ A, or the symbol ∞ if there is no such N. The p-height considers only divisibility properties by the powers of a fixed prime number p. The notion of height admits a refinement so that the p-height becomes an ordinal number. Height plays an important role in Prüfer theorems and also in Ulm's theorem, which describes the classification of certain infinite abelian groups in terms of their Ulm factors or Ulm invariants.

The symbol is known as the number sign, hash, or (in North American usage) pound sign. The symbol has historically been used for a wide range of purposes, including the designation of an ordinal number and as a ligatured abbreviation for pounds avoirdupois – having been derived from the now-rare . Since 2007, widespread usage of the symbol to introduce metadata tags on social media platforms has led to such tags being known as "hashtags", and from that, the symbol itself is sometimes called a hashtag. The symbol is distinguished from similar symbols by its combination of level horizontal strokes and right-tilting vertical strokes.

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).

Ruins of Tiverton Castle, seat of the Earls of Devon Edward de Courtenay, 3rd/11th Earl of Devon (c.1357 - 5 December 1419), known by the epithet the "Blind Earl", was the son of Sir Edward de Courtenay and Emeline Dawnay, and in 1377 succeeded his grandfather, Hugh Courtenay, 10th Earl of Devon, as Earl of Devon. 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.

Here, the basic open sets are the half open intervals [a, b). This topology on R is strictly finer than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from above in the Euclidean topology. This example shows that a set may have many distinct topologies defined on it. If Γ is an ordinal number, then the set Γ = [0, Γ) may be endowed with the order topology generated by the intervals (a, b), [0, b) and (a, Γ) where a and b are elements of Γ. Outer space of a free group Fn consists of the so-called "marked metric graph structures" of volume 1 on Fn.

The church preserved considerable independence in judicial matters, but gave up its old claim that the Norwegian kingdom was a fief under the ultimate authority of the Catholic Church.Sættargjerden i Tunsberg (Store norske leksikon) In cultural terms Magnus continued his father's policy of introducing European courtly culture to Norway. In 1277 he replaced the old Norse titles lendmann and skutilsvein with the European titles baron and riddar (knight), at the same time giving them certain extra privileges and the right to be addressed as lord (herra). Magnus is probably also the first Norwegian king to have named himself using an ordinal number - he called himself "Magnus IV" (he did not count Magnus Haraldsson (II) and Magnus Sigurdsson (IV)).

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.

In mathematics, ordinal logic is a logic associated with an ordinal number by recursively adding elements to a sequence of previous logics.Solomon Feferman, Turing in the Land of O(z) in "The universal Turing machine: a half-century survey" by Rolf Herken 1995 page 111Concise Routledge encyclopedia of philosophy 2000 page 647 The concept was introduced in 1938 by Alan Turing in his PhD dissertation at Princeton in view of Gödel's incompleteness theorems.Alan Turing, Systems of Logic Based on Ordinals Proceedings London Mathematical Society Volumes 2–45, Issue 1, pp. 161–228. While Gödel showed that every system of logic suffers from some form of incompleteness, Turing focused on a method so that from a given system of logic a more complete system may be constructed.

The title of the album is the first to break from the title style of previous Radwimps albums, featuring Radwimps followed by an ordinal number, and followed by a subtitle. This change was intended to show how the mood of the band had changed between this album and Radwimps 4. The title of the album, Altocolony no Teiri, started with the word , which Noda interprets as a key to a problem, as a theorem is in mathematics a statement that is true based on previous mathematical proofs. After listening to the completed album, Noda felt like the album was a theorem explaining himself, that it was a solution to all of his worries and the puzzles before him when he wrote the songs.

Beta is often used to denote a variable in mathematics and physics, where it often has specific meanings for certain applications. In physics a stream of unbound energetic electrons is commonly referred to as beta radiation or beta rays. In regression analysis, symbolizes nonstandardized partial slope coefficients, whereas represents standardized (standard deviation-score form) coefficients; in both cases, the coefficients reflect the change in the criterion Y per one- unit change in the value of the associated predictor X. β is also used in biology, for instance in β-Carotene, a primary source of provitamin A, or the β cells in pancreatic islets, which produce insulin. β is sometimes used as a placeholder for an ordinal number if α is already used.

At the 1815 Battle of Waterloo 14 sergeants of the 40th (the 2nd Somersetshire) Regiment of Foot were killed or wounded while serving in the colour party and at the 1854 Battle of the Alma the colour party of the 21st (Royal North British Fusilier) Regiment of Foot lost 3 officers and 17 sergeants. Colours were first regulated by order of George II in 1747. The recent Jacobite rising of 1745 had prompted the king to set in place army reforms to standardise uniforms, drill and tactics. He was keen to ensure the soldiers' loyalty to the crown rather than the colonel of their regiment (regiments up to this time were known by their colonel's name rather than an ordinal number).

The Spanish era (), sometimes called the era of Caesar, was a calendar era (year numbering system) commonly used in the states of the Iberian Peninsula from the 5th century until the 15th, when it was phased out in favour of the Anno Domini (AD) system. The epoch (start date) of the Spanish era was 1 January 38 BC. To convert an Anno Domini date to the corresponding year in the Spanish era, add 38 to the Anno Domini year, such that Era 941 would be equivalent to AD 903. A date given in the Spanish era always uses the word era followed by a feminine ordinal number (when written out instead of given in Roman numerals). This contrasts with the AD system that uses the masculine anno (year): i.e.

The concept of Bolshevism arose at the Second Congress of the Russian Social Democratic Labour Party (1903) as a result of the split of the party into two factions: supporters of Lenin and the rest.Despite the ordinal number adopted in Soviet historiography, the London Congress was actually a constituent one, since the Minsk Congress had no practical significance One of the main reasons for the split was the question of a party of a new type. In the course of work on the Charter of the Russian Social Democratic Labor Party, Vladimir Lenin and Yuliy Martov proposed two different wordings of the clause on party membership. Lenin – a party member is a citizen who recognizes the program and charter, pays membership fees and works in one of the party organizations.

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.

Any ordinal number can be made into a topological space by endowing it with the order topology (since, being well-ordered, an ordinal is in particular totally ordered): in the absence of indication to the contrary, it is always that order topology that is meant when an ordinal is thought of as a topological space. (Note that if we are willing to accept a proper class as a topological space, then the class of all ordinals is also a topological space for the order topology.) The set of limit points of an ordinal α is precisely the set of limit ordinals less than α. Successor ordinals (and zero) less than α are isolated points in α. In particular, the finite ordinals and ω are discrete topological spaces, and no ordinal beyond that is discrete.

Zero-based numbering is a way of numbering in which the initial element of a sequence is assigned the index 0, rather than the index 1 as is typical in everyday non-mathematical or non-programming circumstances. Under zero-based numbering, the initial element is sometimes termed the zeroth element, rather than the first element; zeroth is a coined ordinal number corresponding to the number zero. In some cases, an object or value that does not (originally) belong to a given sequence, but which could be naturally placed before its initial element, may be termed the zeroth element. There is not wide agreement regarding the correctness of using zero as an ordinal (nor regarding the use of the term zeroth) as it creates ambiguity for all subsequent elements of the sequence when lacking context.

While the 273rd Infantry Division itself initially did not see full redeployment, its subordinate regiments did. The three infantry regiments were deployed on 27 January 1942 at Milowitz military base, were redesignated Grenadier Regiments 544, 545, and 546 on 15 October 1942, and destroyed as part of the 389th Infantry Division at the Battle of Stalingrad between January and February 1943. They were subsequently reassembled under supervision of the 7th Army, sent back to the Eastern Front to fight in the Korsun–Cherkassy Pocket and were eventually trapped at Danzig in 1945. In November 1943, the ordinal number 273 was used for the 273rd Reserve Panzer Division, which was active until March 1944. In April 1945, a second 273rd Infantry Division was deployed as one of the last desperate formations during the final stages of the war.

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".

Any well-ordered set is similar (order-isomorphic) to a unique ordinal number \alpha; in other words, its elements can be indexed in increasing fashion by the ordinals less than \alpha. This applies, in particular, to any set of ordinals: any set of ordinals is naturally indexed by the ordinals less than some \alpha. The same holds, with a slight modification, for classes of ordinals (a collection of ordinals, possibly too large to form a set, defined by some property): any class of ordinals can be indexed by ordinals (and, when the class is unbounded in the class of all ordinals, this puts it in class-bijection with the class of all ordinals). So the \gamma-th element in the class (with the convention that the "0-th" is the smallest, the "1-st" is the next smallest, and so on) can be freely spoken of.

In written languages, an ordinal indicator is a character, or group of characters, following a numeral denoting that it is an ordinal number, rather than a cardinal number. In English orthography, this corresponds to the suffixes -st, -nd, -rd, -th in written ordinals (represented either on the line 1st, 2nd, 3rd, 4th or as superscript, 1st, 2nd, 3rd, 4th). Also commonly encountered are the superscript or superior (and often underlined) masculine ordinal indicator, , and feminine ordinal indicator, , originally from Romance, but via the cultural influence of Italian by the 18th century, widely used in the wider cultural sphere of Western Europe, as in 1º primo and 1ª prima "first, chief; prime quality". The practice of underlined (or doubly underlined) superscripted abbreviations was common in 19th-century writing (not limited to ordinal indicators in particular, and also extant in the numero sign ), and was also found in handwritten English until at least the late 19th century (e.g.

The name Smendes is a hellenization of the Egyptian name Nesbanebdjed ("He of the ram, lord of Mendes"), while the ordinal number distinguishes him from the founder of the 21st Dynasty Smendes I, and from the earlier, namesake High Priest of Amun, Smendes II. A scarcely attested High Priest, he is mainly known for some Nile Level Texts at Karnak where he is called High Priest of Amun and son of king Osorkon: No. 17 (dating to a Year 8 of a deliberately omitted king), No. 18 (Year 13 or 14, king omitted) and No. 19 (Year lost, king omitted).Kitchen, op. cit., § 96; 157. Despite the lack of a conclusive record, it is almost certain that the "king Osorkon" father of Smendes III is Osorkon I: if so, Smendes also was the brother of his two predecessors Iuwelot and Shoshenq C and of the contemporary king Takelot I. Relying on the fact that the previous Nile Level (No.

Arms of Thomas de Courtenay, The Earl of Devon: Arms: Quarterly, 1st and 4th, azure label or three torteaux (for Courtenay); 2nd and 3rd, or a lion rampant azure (for Redvers) Ruins of Tiverton Castle, seat of the Earls of Devon Thomas Courtenay, 6th/14th Earl of Devon (1432 - 3 April 1461), was the eldest son of Thomas de Courtenay, 5th/13th Earl of Devon, by his wife Margaret Beaufort, the daughter of John Beaufort, 1st Earl of Somerset, and Margaret Holland, daughter of Thomas Holland, 2nd Earl of Kent. Through his mother he was a great great-grandson of King Edward III. 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.

In mathematics, specifically in axiomatic set theory, a Hartogs number is a particular kind of ordinal number. In particular, if X is any set, then the Hartogs number of X is the least ordinal α such that there is no injection from α into X. If X can be well-ordered then the cardinal number of α is a minimal cardinal greater than that of X. If X cannot be well-ordered then there cannot be an injection from X to α. However, the cardinal number of α is still a minimal cardinal not less than or equal to the cardinality of X. (If we restrict to cardinal numbers of well-orderable sets then that of α is the smallest that is not not less than or equal to that of X.) The map taking X to α is sometimes called Hartogs's function. This mapping is used to construct the aleph numbers, which are all the cardinal numbers of infinite well- orderable sets.

In combinatorial game theory, the Sprague-Grundy theorem states that every impartial game under the normal play convention is equivalent to a one-heap game of nim, or to an infinite generalization of nim. It can therefore be represented as a natural number, the size of the heap in its equivalent game of nim, as an ordinal number in the infinite generalization, or alternatively as a nimber, the value of that one-heap game in an algebraic system whose addition operation combines multiple heaps to form a single equivalent heap in nim. The Grundy value or nim-value of any impartial game is the unique nimber that the game is equivalent to. In the case of a game whose positions are indexed by the natural numbers (like nim itself, which is indexed by its heap sizes), the sequence of nimbers for successive positions of the game is called the nim-sequence of the game.

In class 332 the hydraulic transmission consists of a clutch and hydrostatic drive, in the 333 of two torque converters For braking the locomotives have a continuous air pressure brake of Knorr design, and in addition, a hand brake for the locomotive which works on the front wheel set. Because the speed of operation of the air compressors which supply the air brake is dependent on the engine speed, one can frequently observe at train stations the locomotive's engines being run (in neutral) at full power to give more braking force. Class 335 The automatic coupling device is visible Locomotives of class 335 differ from class 333 by having indicator lamps which indicate the status of the vehicle to a remote operator as well as an additional box for the remote control on the outside of the drivers cab rear wall Class 335 also have an automatic shunting coupling of a claw type which can also be remoted drawn up out of the way for normal manual screw coupling to take place. In 2001 due to reduced demand for radio controlled locomotives in 24 class 335 locomotives where converted back to class 333; the ordinal number was increased by 500 (thus 335 025 would become 333 525).

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.