In set theory, the successor of an ordinal number α is the smallest ordinal number greater than α.
[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] 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: and for a limit ordinal λ In particular, S(α) = α + 1.
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.