Successor ordinal

In set theory, the successor of an ordinal number α is the smallest ordinal number greater than α. An ordinal number that is a successor is called a successor ordinal. The ordinals 1, 2, and 3 are the first three successor ordinals and the ordinals ω+1, ω+2 and ω+3 are the first three infinite successor ordinals.

Properties

Every ordinal other than 0 is either a successor ordinal or a limit ordinal.

Using von Neumann's ordinal numbers, the successor S of an ordinal number α is given by the formula
Since the ordering on the ordinal numbers is given by α < β if and only if α ∈ β, it is immediate that there is no ordinal number between α and S, and it is also clear that α < S.

The successor operation can be used to define ordinal addition rigorously via transfinite recursion as follows:
and for a limit ordinal λ
In particular,. Multiplication and exponentiation are defined similarly.

The successor points and zero are the isolated points of the class of ordinal numbers, with respect to the order topology.