## Ordinal Numbers

The most common representation of the ordinal numbers, is as follows.

The ordinal 0 is defined to be the empty set

The ordinal 1 is defined to be the set {0} which is equal to

Similarly, the ordinal 2 is the set

The ordinal 3 is the set

The ordinal 4 is the set

Any finite ordinal n is defined to be the set(the successor of an ordinalis defined as the set.

In fact this definition extends naturally to transfinite ordinals. The ordinalis the set consisting of every finite ordinalandis the setis the setand so on.

The ordinalis the set consisting of all finite ordinals and ordinals of the form whereis a finite ordinal so that .

The ordinalis the first uncountable ordinal, and is the set of all countable ordinals.