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.
Every ordinal other than 0 is either a successor ordinal or a limit ordinal.[1]
Using von Neumann's ordinal numbers (the standard model of the ordinals used in set theory), the successor S(α) of an ordinal number α is given by the formula[1]
S(\alpha)=\alpha\cup\{\alpha\}.
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(α), and it is also clear that α < S(α).
The successor operation can be used to define ordinal addition rigorously via transfinite recursion as follows:
\alpha+0=\alpha
\alpha+S(\beta)=S(\alpha+\beta)
and for a limit ordinal λ
\alpha+λ=cup\beta(\alpha+\beta)
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.[2]