ordinal number

Noun

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Noun

**ordinal number** (*plural* ordinal numbers)

- (
*grammar*) 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.

- (
*arithmetic*) A natural number used to denote position in a sequence. - (
*set theory*) 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.

- For not only do the antinomies a) to e) disappear when we admit as elements of sets only such sets,
**1960**[D. Van Nostrand], Paul R. Halmos,*Naive Set Theory (book)*, 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.

- Is there a set that consists exactly of all the
**2009**, Marek Kuczma, Attila Gilányi (editor),*An Introduction to the Theory of Functional Equations and Inequalities*, Springer (Birkhäuser), 2nd Edition, page 10 ↗,- If \alpha is an
**ordinal number**, then by definition any two well-ordered sets of type \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**\alpha we may assign a cardinal number, the common cardinality of all well-ordered sets of type \alpha.

- If \alpha is an

- (
*grammar*) ordinal, ordinal numeral - (
*arithmetic*) ordinal - (
*order theory*) ordinal

- (
*grammar*) cardinal, cardinal number, cardinal numeral

- French: nombre ordinal (
*pl.*nombres ordinaux) - German: Ordnungszahl, Ordnungszahlwort; Ordinale, Ordinalzahl
- Portuguese: número ordinal
- Russian: поря́дковое числи́тельное

- Russian: поря́дковый но́мер

This text is extracted from the Wiktionary and it is available under the CC BY-SA 3.0 license | Terms and conditions | Privacy policy 0.005