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,
| symbol | elements | description |
| 0 |  | empty set |
| 1 |  | set of one element |
| 2 |  | set of two elements |
| 3 |  | set of three elements |
 | ||
 |  | set of all finite ordinals |
 |  | |
 | ||
 | set of all countable ordinals | |
 | ||
 | set of all ordinals of cardinality
less than  | |
 | ||
 | set of all ordinals of cardinality
less than  | |
 |
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.
