Ordinal Number

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, ..., i.e., the integers one less than the corresponding nonnegative integers.

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 the ordinal number of the whole numbers. Conway and Guy (1996) denote it with the notation .

From the definition of ordinal comparison, it follows that the ordinal numbers are a well ordered set. 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 countable and ordinals set all finite ordinals and ordinals for all nonnegative integers

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 numbers have some other rather peculiar properties. The sum of two ordinal numbers can take on two different values, the sum of three can take on five values. 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 sum of ordinals has either or possible answers for (Conway and Guy 1996).

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

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.