This article is about the mathematical concept. For number words denoting a position in a sequence ("first", "second", "third", etc.), see Ordinal numeral.
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Representation of the ordinal numbers up to ωω. One turn of the spiral corresponds to the mapping . Since has as least fixed point, larger ordinal numbers cannot be represented.
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite sets.[1]
A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally as linearly ordered labels that include the natural numbers and have the property that every set of ordinals has a least element (this is needed for giving a meaning to "the least unused element").[2] This more general definition allows us to define an ordinal number (omega) that is greater than every natural number, along with ordinal numbers , , etc., which are even greater than .
A linear order such that every non-empty subset has a least element is called a well-order. The axiom of choice implies that every set can be well-ordered, and given two well-ordered sets, one is isomorphic to an initial segment of the other. So ordinal numbers exist and are essentially unique.
Ordinal numbers are distinct from cardinal numbers, which measure the size of sets. Although the distinction between ordinals and cardinals is not always apparent on finite sets (one can go from one to the other just by counting labels), they are very different in the infinite case, where different infinite ordinals can correspond to sets having the same cardinal. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated, although none of these operations are commutative.
Ordinals were introduced by Georg Cantor in 1883[3] in order to accommodate infinite sequences and classify derived sets, which he had previously introduced in 1872 while studying the uniqueness of trigonometric series.[4]
^"Ordinal Number - Examples and Definition of Ordinal Number". Literary Devices. 2017-05-21. Retrieved 2021-08-31.
^Sterling, Kristin (2007-09-01). Ordinal Numbers. LernerClassroom. ISBN 978-0-8225-8846-7.
^Thorough introductions are given by Levy 1979 and Jech 2003.
^Hallett, Michael (1979), "Towards a theory of mathematical research programmes. I", The British Journal for the Philosophy of Science, 30 (1): 1–25, doi:10.1093/bjps/30.1.1, MR 0532548. See the footnote on p. 12.
In set theory, an ordinalnumber, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite...
have the same ordinalnumber. The first ordinalnumber that is not a natural number is expressed as ω; this is also the ordinalnumber of the set of natural...
In linguistics, ordinal numerals or ordinalnumber words are words representing position or rank in a sequential order; the order may be of size, importance...
Continuing in this manner, it is possible to define a cardinal number ℵα for every ordinalnumber α, as described below. The concept and notation are due to...
into Cardinal and Ordinal Numbers (1958, 2nd ed. 1965). Any finite natural number can be used in at least two ways: as an ordinal and as a cardinal....
arbitrary numerical scale Ordinal date, a simple form of expressing a date using only the year and the day number within that year Ordinal Priority Approach,...
An ordinal date is a calendar date typically consisting of a year and an ordinalnumber, ranging between 1 and 366 (starting on January 1), representing...
an ordinal indicator is a character, or group of characters, following a numeral denoting that it is an ordinalnumber, rather than a cardinal number. In...
number sign, hash, or pound sign. The symbol has historically been used for a wide range of purposes including the designation of an ordinalnumber and...
limit ordinal is an ordinalnumber that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less...
the von Neumann representation of ordinals. Larger ordinal fixed points of the exponential map are indexed by ordinal subscripts, resulting in ε 1 , ε...
uncountable ordinal, traditionally denoted by ω 1 {\displaystyle \omega _{1}} or sometimes by Ω {\displaystyle \Omega } , is the smallest ordinalnumber that...
numbers are indexed by ordinal numbers. If the axiom of choice is true, this transfinite sequence includes every cardinal number. If the axiom of choice...
Ordinal data is a categorical, statistical data type where the variables have natural, ordered categories and the distances between the categories are...
to distinguish monarchs. An ordinal is the number placed after a monarch's regnal name to differentiate between a number of kings, queens or princes reigning...
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation...
The surreal number ω − 1 is not an ordinal; the ordinal ω is not the successor of any ordinal. This is a surreal number with birthday ω+1, which is labeled...
mathematics Japanese punctuation Korean punctuation Ordinal indicator – Character(s) following an ordinalnumber (used of the style 1st, 2nd, 3rd, 4th or as superscript...
smallest ordinalnumber greater than the ranks of all members of the set. In particular, the rank of the empty set is zero, and every ordinal has a rank...
when the number follows the noun it determines, as in ora doisprezece "12 o'clock" or clasa a doisprezecea ("12th grade", see below for ordinal numbers);...
thereby assigning it 0 elements. Also in set theory, 0 is the lowest ordinalnumber, corresponding to the empty set viewed as a well-ordered set. In order...
constant. In set theory, the first infinite ordinalnumber, ω In set theory, the first uncountable ordinalnumber, ω1 or Ω As part of logo or trademark: The...
registry number, three digits (SSS) as a combination of the citizen's sex and ordinalnumber of birth, and one digit (C) as a control number. The two...
Hartogs number is an ordinalnumber associated with a set. In particular, if X is any set, then the Hartogs number of X is the least ordinal α such that...
described by an ordinalnumber. For instance, 3 is the ordinalnumber of the set {0, 1, 2} with the usual order 0 < 1 < 2; and ω is the ordinalnumber of the set...
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