# ordinal number

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### Noun

ordinal number (plural ordinal numbers)

1. (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.
2. (arithmetic) A natural number used to denote position in a sequence.
In the expression a3, the "3" is an ordinal number.
3. (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.
• 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.
• 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 $\alpha$ .

#### Translations

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#### Usage notes

On ordinal number usage:

eleventh day
• If an ordinal is followed by a plural noun, the two-word phrase refers to a set of items described by the phrase in singular. For example second homes refers to a set of homes each of which is considered a "second home."
two fifths , $\textstyle {\frac {2}{5}}$ • Ordinal numbers are used in exponents, where generally construed as adjectives preceding power to which a base is raised:
two to the minus twenty-first power , $2^{-21}$ six to the third , $6^{3}$ • Ordinal numbers are generally considered to be ordered from high to low , so that first place is considered highest , and fifth is lower than second. Degree is an exception.
• Ordinal numbers corresponding to numbers higher than 20 use cardinal numbers for all the places preceding the final ordinal part:
twenty-first or 21st , occasionally XXI
one hundred fifteenth or 115th , occasionally CXV
thirty-three thousandth or 33,000th