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In common usage, an ordinal number is an adjective which describes the numerical position of an object, e.g., first, second, third, etc.

In formal set theory, an ordinal number (sometimes simply called an "ordinal" for short) is one of the numbers in Georg Cantor's extension of the whole numbers. An ordinal number is defined as the order type of a well ordered set (Dauben 1990, p. 199; Moore 1982, p. 52; Suppes 1972, p. 129). Finite ordinal numbers are commonly denoted using arabic numerals, while transfinite ordinals are denoted using lower case Greek letters.

It is easy to see that every finite totally ordered set is well ordered. Any two totally ordered sets with elements (for a nonnegative integer) are order isomorphic, and therefore have the same order type (which is also an ordinal number). The ordinals for finite sets are denoted 0, 1, 2, 3, ..., with the finite ordinal corresponding to a set of elements.

The first transfinite ordinal, denoted , is the order type of the set of nonnegative integers (Dauben 1990, p. 152; Moore 1982, p. viii; Rubin 1967, pp. 86 and 177; Suppes 1972, p. 128). This is the "smallest" of Cantor's transfinite numbers, defined to be the smallest ordinal number greater than every finite ordinal. Conway and Guy (1996) denote it with the notation .

From the definition of ordinal comparison, it follows that every set of ordinal numbers is well ordered. In order of increasing size, the ordinal numbers are 0, 1, 2, ..., , , , ..., , , .... The notation of ordinal numbers can be a bit counterintuitive, e.g., even though , . The cardinal number of the set of countable ordinal numbers is denoted (aleph-1).

If is a well ordered set with ordinal number , then the set of all ordinals is order isomorphic to . This provides the motivation to define an ordinal as the set of all ordinals less than itself. John von Neumann defined a set to be an ordinal number iff

1. If is a member of , then is a proper subset of

2. If and are members of then one of the following is true: , is a member of , or is a member of .

3. If is a nonempty proper subset of , then there exists a member of such that the intersection is empty.

(Rubin 1967, p. 176; Ciesielski 1997, p. 44). This is the standard representation of ordinals. In this representation,

Rubin (1967, p. 272) provides a nice definition of the ordinals.

Since for any ordinal , the union is a bigger ordinal , there is no largest ordinal, and the class of all ordinals is therefore a proper class (as shown by the Burali-Forti paradox).

Ordinal addition is generally not commutative. The maximum number of distinct sums obtained by permuting a fixed collection of two ordinals is 2, and for three ordinals it is 5. For , 3, ..., the first few terms of this sequence are 2, 5, 13, 33, 81, 193, 449, , , , , , ..., namely 2, 5, 13, 33, 81, 193, 449, 1089, 2673, 6561, 15633, 37249, ... (Conway and Guy 1996, OEIS A005348). The maximum number of distinct sums of ordinals is of the form or for nonnegative integers and (Conway and Guy 1996).

For a positive integer , is the same as , but is equal to . is larger than any number of the form , is larger than , and so on.

There exist ordinal numbers which cannot be constructed from smaller ones by finite additions, multiplications, and exponentiations. These ordinals obey Cantor's equation. The first such ordinal is

The next is

where and . Then follow , , ..., , , ..., , ..., , , ..., , , ..., , ..., , ..., , ..., , ..., , ..., , ..., , ..., , ..., , ..., , ... (Conway and Guy 1996).

Ordinal addition, ordinal multiplication, and ordinal exponentiation can all be defined. Although these definitions also work perfectly well for order types, this does not seem to be commonly done. There are two methods commonly used to define operations on the ordinals: one is using sets, and the other is inductively.

Ordinal numbers are represented in the Wolfram Language by Ordinal.

See also

Aleph-1, Axiom of Choice, Burali-Forti Paradox, Cantor's Equation, Cardinal Number, Initial Ordinal, Order Statistic, Order Type, Power Set, Surreal Number, Well Ordered Set

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References

Cantor, G. Über unendliche, lineare Punktmannigfältigkeiten, Arbeiten zur Mengenlehre aus dem Jahren 1872-1884. Leipzig, Germany: Teubner-Archiv zur Mathematik, 1884.Ciesielski, K. Set Theory for the Working Mathematician. Cambridge, England: Cambridge University Press, 1997.Conway, J. H. and Guy, R. K. "Cantor's Ordinal Numbers." In The Book of Numbers. New York: Springer-Verlag, pp. 266-267 and 274, 1996.Dauben, J. W. Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton, NJ: Princeton University Press, 1990.Moore, G. H. Zermelo's Axiom of Choice: Its Origin, Development, and Influence. New York: Springer-Verlag, 1982.Numberphile. "Ordinal Numbers." Oct. 22, 2025. https://www.youtube.com/watch?v=47qEMTMKRdA.Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967.Sloane, N. J. A. Sequence A005348/M1435 in "The On-Line Encyclopedia of Integer Sequences."Suppes, P. Axiomatic Set Theory. New York: Dover, 1972.

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Ordinal Number

Cite this as:

Weisstein, Eric W. "Ordinal Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OrdinalNumber.html

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