ordinal number en · NOUN
Pronunciation
- (Southern-England) audio
Meanings
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A word that expresses the relative position of an item in a sequence.
First, second and third are the ordinal numbers corresponding to one, two and three.
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A natural number used to denote position in a sequence.
In the expression a₃, the "3" is an ordinal number.
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Such a number generalised to correspond to any cardinal number (the size of some set); formally, the order type of some well-ordered set of some cardinality a, which represents an equivalence class of well-ordered sets (exactly those of cardinality a) under the equivalence relation "existence of an order-preserving bijection".
1950, Frederick Bagemihl (translator), Erich Kamke, Theory of Sets, Dover (Dover Phoenix), 2006, page 137, For not only do the antinomies a) to e) disappear when we admit as elements of sets only such sets, ordinal numbers, and cardinal numbers as are bounded above by a fixed cardinal number, but we see also that paradoxes always arise if we collect into a set any sets, cardinal numbers, or ordinal numbers which are not bounded above by a fixed cardinal number.
1960 [D. Van Nostrand], Paul R. Halmos, Naive Set Theory, 2017, Dover, Republication, page 80, Is there a set that consists exactly of all the ordinal numbers? It is easy to see that the answer must be no. If there were such a set, then we could form the supremum of all ordinal numbers. That supremum would be an ordinal number greater than or equal to every ordinal number. Since, however, for each ordinal number there exists a strictly greater one (for example, its successor), this is impossible; it makes no sense to speak of the "set" of all ordinals.
If #92;alpha is an ordinal number, then by definition any two well-ordered sets of type #92;alpha are similar, i.e., there exists a one-to-one mapping from one set to the other. Consequently these sets have the same cardinality. Consequently to any ordinal number #92;alpha we may assign a cardinal number, the common cardinality of all well-ordered sets of type #92;alpha.
2009, Marek Kuczma, edited by Attila Gilányi, An Introduction to the Theory of Functional Equations and Inequalities, 2nd edition, Springer (Birkhäuser), page 10:
Forms
| Spelling | Features | Labels | Source |
|---|---|---|---|
| ordinal numbers | Number=Plur | lexicographic |
Antonyms
Coordinates
distributive number · multiplicative number multiplier · cardinal number
Relateds
numeral · order type · zeroth
Synonyms
ordinal numeral · ordinal · ordinal · ordinal
Translations (59)
ja 順序数 (generalized kind of number to denote the length of a well-order on a set) · bg ординал (generalized kind of number to denote the length of a well-order on a set) · tl bilang na panunod (arithmetic: number used to denote position in a sequence) · fi järjestyslukusana (grammar: word used to denote relative position in a sequence) · de Ordinale (grammar: word used to denote relative position in a sequence) · fo raðtal (grammar: word used to denote relative position in a sequence) · cmn 序數 /序数 (grammar: word used to denote relative position in a sequence) · cs řadová číslovka (grammar: word used to denote relative position in a sequence) · fi järjestysluku (arithmetic: number used to denote position in a sequence) · el τακτικό αριθμητικό (grammar: word used to denote relative position in a sequence) · pt número ordinal (grammar: word used to denote relative position in a sequence) · pl liczba porządkowa (generalized kind of number to denote the length of a well-order on a set) · fi ordinaalilukusana (grammar: word used to denote relative position in a sequence) · cs ordinální číslo (generalized kind of number to denote the length of a well-order on a set) · cmn 序數詞 /序数词 (grammar: word used to denote relative position in a sequence) · pl liczba porządkowa (grammar: word used to denote relative position in a sequence) · nb ordenstall (grammar: word used to denote relative position in a sequence) · nl ranggetal (generalized kind of number to denote the length of a well-order on a set) · ka რიგითი რიცხვითი სახელი (grammar: word used to denote relative position in a sequence) · fr nombre ordinal (pl. nombres ordinaux) (grammar: word used to denote relative position in a sequence) · pl liczebnik porządkowy (grammar: word used to denote relative position in a sequence) · sv ordningstal (grammar: word used to denote relative position in a sequence) · hu sorszámnév (grammar: word used to denote relative position in a sequence) · da ordenstal (grammar: word used to denote relative position in a sequence) · nl rangtelwoord (grammar: word used to denote relative position in a sequence) · is raðtala (grammar: word used to denote relative position in a sequence) · la numerus ordinālis (grammar: word used to denote relative position in a sequence) · cmn 序數 /序数 (arithmetic: number used to denote position in a sequence) · ja 序数 (grammar: word used to denote relative position in a sequence) · cmn 順序數 /顺序数 (grammar: word used to denote relative position in a sequence) · cy trefnolyn (grammar: word used to denote relative position in a sequence) · cy trefnolyn (generalized kind of number to denote the length of a well-order on a set) · crh sıra sayısı (grammar: word used to denote relative position in a sequence) · pl liczba porządkowa (arithmetic: number used to denote position in a sequence) · hu rendszám (generalized kind of number to denote the length of a well-order on a set) · cy rhif trefnol (grammar: word used to denote relative position in a sequence) · ru но́мер (arithmetic: number used to denote position in a sequence) · mi tau raupapa (arithmetic: number used to denote position in a sequence) · bg пореден номер (arithmetic: number used to denote position in a sequence) · ko 서수 (grammar: word used to denote relative position in a sequence) · gl número ordinal (grammar: word used to denote relative position in a sequence) · de Ordnungszahlwort (grammar: word used to denote relative position in a sequence) · cy rhif trefnol (generalized kind of number to denote the length of a well-order on a set) · bg числително редно (grammar: word used to denote relative position in a sequence) · fi ordinaaliluku (arithmetic: number used to denote position in a sequence) · de Ordinalzahl (grammar: word used to denote relative position in a sequence) · cy rhif trefnol (arithmetic: number used to denote position in a sequence) · da ordinaltal (grammar: word used to denote relative position in a sequence) · cy trefnolyn (arithmetic: number used to denote position in a sequence) · de Ordnungszahl (grammar: word used to denote relative position in a sequence) · ka რიგობითი რიცხვითი სახელი (grammar: word used to denote relative position in a sequence) · fi ordinaali (generalized kind of number to denote the length of a well-order on a set) · ja 序数詞 (grammar: word used to denote relative position in a sequence) · ca nombre ordinal (grammar: word used to denote relative position in a sequence) · hi कर्मवाचक संख्या (grammar: word used to denote relative position in a sequence) · sh redni broj (grammar: word used to denote relative position in a sequence) · ko 서수사 (grammar: word used to denote relative position in a sequence) · ru поря́дковый но́мер (arithmetic: number used to denote position in a sequence) · ru поря́дковое числи́тельное (grammar: word used to denote relative position in a sequence)