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 ordinal
is defined as the set
.
In fact this definition extends naturally to transfinite ordinals. The ordinal
is the set consisting of every finite ordinal
and
is the set
is the set
and so on.
The ordinal
is the set consisting of all finite ordinals and ordinals of the form
where
is a finite ordinal so that
.
The ordinal
is the first uncountable ordinal, and is the set of all countable ordinals.